Research ArticleA Blast-Resistant Method Based on Wave Converters withSpring Oscillator for Underground Structures

Yu Zhang1 Yuanxue Liu1 RunzeWu1 Jichang Zhao1 Ming Hu1 and Yizhong Tan2

1Chongqing Key Laboratory of Geomechanics amp Geoenvironmental Protection Logistical Engineering University Chongqing China2PLA Engineering College Xuzhou China

Correspondence should be addressed to Yuanxue Liu 357202780qqcom

Received 20 January 2017 Accepted 12 April 2017 Published 10 May 2017

Academic Editor Nuno M Maia

Copyright copy 2017 Yu Zhang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Researches on blast-resistant measures for underground structures such as tunnels and underground shopping malls are of greatimportance for their significant role in economic and social development In this paper a new blast-resistantmethod based onwaveconverters with spring oscillator for underground structures was put forward so as to convert the shock wave with high frequencyand high peak pressure to the periodic stress wave with low frequency and low peak pressure The conception and calculationprocess of this new method were introduced The mechanical characteristics and motion evolution law of wave converters werededuced theoretically Based on the theoretical deduction results and finite difference software FLAC3D the dynamic responses ofthe new blast-resistant structure and the traditional one were both calculated Results showed that after the deployment of waveconverters the peak absolute values of the bending moment shear force and axial force of the structure decreased generally whichverified the good blast-resistant effect of the new blast-resistant method

1 Introduction

Tunnels subway stations underground shopping malls andso forth play an important role in the economic and socialdevelopment as well as the personal and property safety Inrecent years there have been a variety of researches on theblast-resistant measures for underground structures mainlyfocused on the design and optimization on structures andmaterials

Blast-resistant measures based on the structure optimiza-tion mainly contain the increase of stiffness or adoption ofstructures good for the reflection diffraction and scatteringof the stress wave Usually the arch structures and structurescontaining holes have better performance of wave dissipationthan rectangular structures or structures without holes sothese kinds of structures have attracted many attentions ofscholars [1ndash3] A good blast-resistant performance is alsoembodied in the plate-foam composite structure [4] box-shaped steel plate-reinforced concrete composite structure[5] carbon fiber reinforced composite structure [6 7] andprestressed structure [8] Via the explosion model testsYang et al studied the antiexplosion effect of prestressed

cable reinforced caverns The results showed that prestressedstructures are good for antiexplosion [9] Kobielak et alexperimentally investigated the influence of an attenuationbarrier on soil stresses and pressures acting on a buried silocaused by underground explosions at different distances Testresults verified the blast-resistant effectiveness caused by thebarrier composed of the cylindrical tubes [10] The dynamicresponse of a buried silo caused by underground explosionwas also studied experimentally via the measured pressures[11]

Blast-resistant measures based on the material optimiza-tion are mainly concentrated on the development of porousor lightweight materials of low stiffness and the materials aredeveloping gradually from the traditional inorganic porousmaterials or lightweight materials to the polymer materialsand porous metal materials currently Materials such asthe rigid polyurethane foam polypropylene fiber concreterubber concrete foam concrete foamed aluminium andsteel fiber reinforced concrete are good choices for the blast-resistant materials Yakushin et al investigated the propertiesof low-density rigid polyurethane foams with hollow glass

HindawiShock and VibrationVolume 2017 Article ID 2014726 13 pageshttpsdoiorg10115520172014726

2 Shock and Vibration

microspheres The tension and compression properties inrelation to the content of microspheres were determined intheir work [12] Alhozaimy et al found that the flexuraltoughness and impact resistance showed an increase inthe presence of polypropylene fibers [13] The mechanicalproperties of concrete containing tire-rubber particles werestudied by Khaloo et al [14] They found that unlike plainconcrete the failure state in rubberized concrete occurredgently and uniformly and did not cause any separation inthe specimen In fact this property is good for dynamicprotection Hernandez-Olivares and Barluenga also studiedthe fire performance of the rubber-filled concrete [15] Limet al studied the compressive splitting tensile and flexuralstrengths of lightweight foamed concrete [16] The resultshave shown that the foamed concrete is a good choice fordynamic protection

In the blast-resistant methods of the traditional structureoptimization the construction process is usually complexand sometimes the function of structures may even beaffected The shock wave mainly consists of high frequencycomponents In the blast-resistant methods of porous orlightweight materials the materials are easy to get damagedunrecoverably and have large deformation under the blastingload because of the low elastic modulus and the existenceof the holes Thereby the overall stability of undergroundstructure and surrounding rock may be affected by the largedeformation

In order to improve the traditional antiknock methodsa new blast-resistant method based on wave converterswith spring oscillator for underground structures is putforward in this paper The new method mainly consists ofan array of wave converters and a distribution layer Firstlythe conception of the new method is introduced includingthe formation of the wave converter and distribution layerSecondly the calculation process of dynamic responses forunderground structures adopting the new blast-resistantmethod is presented Thirdly the mechanical characteristicsand motion evolution law of the wave converter are derivedincluding the static constitutive relation of the wave con-verter dynamic response partitioning of the wave converterdifferential equation of motion for the spring oscillatordisplacement transfer coefficient of the wave converter andthe stress inversion of the wave converter A case study isalso conducted to verify the applicability and rationality ofthe newmethod by comparing with the traditional structure

2 Conception of New Blast-Resistant Method

The new method mainly consists of an array of wave con-verters and a distribution layer shown in Figure 1 The waveconverter includes 2 shells containing a length adjusting rodand a spring oscillator comprised of springs and amass blockshown in Figure 2 The initial length of the wave convertercan be adjusted via the adjustment of the length adjustingrod which can control the prestress of the compressionsprings Via the adjustment of the initial converter lengththe wave converter can also be easily installed under differentreserve space between the rock and tunnel roof The waveconverter is compressible when the load on the top of

Distribution layer

Underground structure

Rock

Wave converters

Figure 1 Schematic of the new blast-resistant method

Mass block

Upper spring

Lower spring

Length adjusting rod

Upper shell

Lower shell

Spring oscillator

Figure 2 Schematic of the wave converter

the converter exceeds the spring prestress The distributionlayer is composed of the material with relatively low waveimpedance shown in Figure 1 Multiple reflections caused bythe periodic stress wave can occur in the distribution layerwhich results in the further energy dissipation

The new blast-resistant method combines such mecha-nisms as the spring deformation inertia and periodic vibra-tion of the mass block to provide the resistance against thedynamic load The self-support capacity of the surroundingrock can also be fully utilized Via the wave converter theshock wave with high frequency and high peak pressure canbe transformed to the periodic stresswavewith low frequencyand low peak pressure Thereby the shock wave is dispersedand materials under the converter can be prevented fromcrushing Under the blasting load the deformation process ofthe wave converter can be divided into such 3 periods as rapidloading stage rapid unloading stage and slow unloadingstage

The above 3 stages are determined by the relative displace-ment Δ119906119894(119905) between the top and bottom of wave converterUnder the impact loads propagating in the rock the typicalcurve of Δ119906119894(119905) with time for the wave converter is shownin Figure 3 As is shown in Figure 3 the rapid loading stagecorresponds to the sharp increasing period and the rapidunloading stage corresponds to the fast decreasing period

Shock and Vibration 3

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

Δui(t)

002 004 006 008 010 012 014 016 018000Dynamic time (s)

Rapid loadingstage Rapid unloading

stageSlow unloading stage

Figure 3 Typical curve of Δ119906119894(119905) with time for the wave converter

Air

Rock

Distribution layer

Underground structure

Monitoring position for displacements

Rock surface at the wave converter top

Distribution layer surface at the wave converter bottom

Initial pressure Pin caused by the wave converter

Figure 4 Schematic of model 1

while the slow unloading stage corresponds to the slowchange period

3 Calculation Process of DynamicResponses for Structures with the NewBlast-Resistant Method

(1) Establish numerical models and acquire the dataneeded for the calculation of the wave converterrsquosdisplacements in the rapid loading stage and rapidunloading stage Model 1 without wave converters forthe finite element analysis is set up shown in Figure 4In model 1 the distribution layer underground struc-ture and rock are established The initial pressurecaused by the wave converter on the distribution layersurface (at the wave converter bottom) is applied asthe lower spring force divided by the cross-sectional

area of the wave converter Then the explosion posi-tion and blasting load should be applied After thatunder the blasting load the finite element analysismethod is used to calculate the vertical displacement-time curves 1199060119894(119905) and 1199062119894(119905) of the rock surface atthe wave converter top and the distribution layer sur-face at the wave converter bottom respectively Themonitoring positions for displacements are suggestedto adopt the tops and bottoms of 9 wave convertersalong the width direction of the structure shown inFigure 4 Then model 2 of the ground without anyconstruction such as the structure distribution layerand wave converters is built up shown in Figure 5The numerical analysis based on model 2 under thesame explosion condition as model 1 is conducted toobtain the vertical rock pressure-time curve1198751198600(119905) forthe corresponding monitoring positions of the rocksurface in model 1 shown in Figure 5

(2) Solve the differential equation of motion for thespring oscillator and obtain the law of motion inthe rapid loading stage and rapid unloading stage Inmodel 1 the vertical displacement-time curve1199060119894(119905) ofthe rock surface at the wave converter top multipliedby a displacement transfer coefficient 119870119894 (0 lt 119870119894 lt1) is considered as the vertical displacement-timecurve 1199061119894(119905) of the wave converter top in the rapidloading stage and rapid unloading stage of the truesituation The true situation refers to the real under-ground structure with an array of wave convertersand a distribution layer The vertical displacement-time curve 1199062119894(119905) of the distribution layer surface atthe wave converter bottom in model 1 is considereddirectly as the vertical displacement-time curve of thewave converter bottom in the rapid loading stage andrapid unloading stage of the true situation In the truesituation the difference between 1199061119894(119905) and 1199062119894(119905) isthe relative displacement Δ119906119894(119905) between the top andbottom of the wave converter The wave converter isin the rapid loading stage before the relative displace-ment reaches the maximum while it is in the rapidunloading stage during the sharp decrease periodafter the relative displacement reaches the maximumTaking the displacements of the wave converter topand bottom as the boundary condition the differen-tial equation ofmotion for the spring oscillator can besolved to obtain the law ofmotion in the rapid loadingstage and rapid unloading stage Then the stress-timecurve of thewave converter top and bottom in above 2stages can be gotten

(3) Solve the differential equation of motion for thespring oscillator and obtain the law of motion inthe slow unloading stage According to step (2) thelength of the wave converter the displacement (orposition coordinate) and velocity of the oscillator atthe end of the rapid unloading step can be gottenThen the differential equation ofmotion for the springoscillator in the slow unloading stage can be solvedIn the slow unloading stage the length of the wave

4 Shock and Vibration

Rock

Rock

RockRock

Monitoring position for the vertical stress

Rock surface at the wave converter top in model 1

Rock

Figure 5 Schematic of model 2

converter can be deemed as a constant In that casethe calculation results of the dynamic response are alittle dangerous whichwould be safer for the structuredesign After that the stress-time curve of the waveconverter top and bottom in the slow unloading stagecan be gotten

(4) Calculate the dynamic response of the undergroundstructure Firstly the initial pressure 119875in caused bythe wave converter on the distribution layer surfacein model 1 should be deleted Then the stress-timecurves of the wave converter top and bottom in threestages are applied on the rock surface and distributionlayer surface instead of the wave converters Afterthat the blasting load is applied and the dynamiccalculation is conducted to get the dynamic responseof the structure

4 Mechanical Characteristics and MotionEvolution Law of Wave Converters

41 Static Constitutive Relation of the Wave Converter 1198970 isthe initial length of the wave converter while 119886 is the shellthickness So the initial clear length of the wave convertercan be written as 1198971198990 = 1198970 minus 2119886 and 1198971198990 is shown in Figure 6In Figure 6 the dashed line is the position under the staticequilibrium and the solid line is the position at any timeunder the dynamic load 1198961 is the stiffness coefficient of theupper spring whose length in free state is 11989710 Δ11990910 is theinitial amount of compression for upper spring 1198962 is thestiffness coefficient of the lower spring and 11989720 is its length infree stateΔ11990920 is the initial amount of compression for lowerspring 119898 and 1198973 are the mass and height of the mass blockrespectively 119875119860(119905) is the vertical rock pressure on the waveconverter top in the true situation while 1198751198600(119905) is the verticalrock pressure at the same location inmodel 2When the wave

d

PA(t)

o A

Au1i(t)

zi(t)

ln0 + u2i(t)

z

B

B

PB(t)

mi(t)ai(t)

k2

k1

l2 = l20 minus Δx20

l1 = l10 minus Δx10

l3ln0

Figure 6 Deformation process of the wave converter

converter is installed with the initial length 1198970 under staticequilibrium the geometric equation is established yielding

1198971198990 = 1198971 + 1198972 + 1198973 = 11989710 + 11989720 + 1198973 minus Δ11990910 minus Δ11990920 (1)

As for the mass block the balance equation is

119898119892 + 1198961Δ11990910 = 1198962Δ11990920 (2)

Combining (1) and (2) the solutions are

Δ11990910 = 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

Δ11990920 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 1198971198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (3)

1198651 and 1198652 are the force of upper and lower springsrespectively 119904 is the cross-sectional area of thewave converter119889 is the cross-sectional length If the weight of the waveconverterrsquos shells is ignored 1198651 can be expressed as

1198651 (119905) = 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) 119904119875119860 (119905) ge 1198961Δ11990910 (4)

1198652 can be expressed as

1198652 (119905) = 119898119892 + 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) + 119898119892 119904119875119860 (119905) ge 1198961Δ11990910 (5)

Shock and Vibration 5

So the static constitutive relation of the wave converteryields

If 119904119875119860 (119905) lt 1198961Δ11990910Δ119897 = 0 (6)

If 119904119875119860 (119905) ge 1198961Δ11990910Δ119897 = (1198961 + 1198962) 119904119875119860 (119905)11989611198962 + [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]1198962 (1198961 + 1198962) (7)

42 Dynamic Response Partitioning of the Wave ConverterThe rapid loading stage rapid unloading stage and slowunloading stage correspond to the rapid compression stagerapid recovery stage and slow recovery stage respectively Inthe slow recovery stage the length of the wave converter canbe deemed as a constantThe demarcation point of stage 1 andstage 2 is that the relative displacement reaches themaximumThe duration time of stage 2 can be determined by the relativedisplacement-time curve When the rock masses above thestructure are in a wide range of elasticity state the rapidunloading stage can not be ignored but if the rockmasses arein a wide range of plastic state the rapid unloading stage canbe ignored

The computing time of stage 3 is advisable for 1 or 2vibration periods Via a large amount of computations it isconcluded that the computing time of stage 3 can be taken as1 vibration period if stage 2 can not be ignored otherwise itcan be taken as 2 vibration periods

43 Differential Equation of Motion for the Spring Oscillator

431 Differential Equation of Motion in Rapid Loading Stageand Rapid Unloading Stage 119873 is the total number of waveconverters along the width direction of the structure and 119894 isthe serial number of the wave converter For wave converter119894 1199060119894(119905) is the vertical displacement of the rock surface at thewave converter top in model 1 and 1199062119894(119905) (119894 = 1 2 119899)is the vertical displacement of the wave converter bottom in

rapid loading stage and rapid unloading stage of the true situ-ation 1199061119894(119905) is the vertical displacement of the wave convertertop in rapid loading stage and rapid unloading stage of thetrue situation which can be expressed as 1199061119894(119905) = 1198701198941199060119894(119905)119870119894is the displacement transfer coefficient of the wave converterΔ119906119894(119905) is the relative displacement between the top and bot-tom of the wave converter which can be written as Δ119906119894(119905) =1199061119894(119905) minus 1199062119894(119905) At the end of the rapid unloading stage is theinitial state of the slow unloading stage which can be deemedas the fixed-length vibration shown in Figure 7 In Figure 7the dashed line is the position under the static equilibrium inthe fixed-length vibration and the solid line is the positionat the end of the rapid unloading stage

As is shown in Figure 6 V119894(119905) 119886119894(119905) and 119911119894(119905) are oscillatorvelocity oscillator acceleration and position coordinate at thetime of 119905 respectively yielding

V119894 (119905) = 119889119911119894 (119905)119889119905 (8)

The length of the upper spring is

1198971119894 (119905) = 119911119894 (119905) minus 1199061119894 (119905) minus 11989732 (9)

The amount of the spring compression is

Δ1199091119894 (119905) = 11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732 (10)

The length of the lower spring is

1198972119894 (119905) = 1198971198990 + 1199062119894 (119905) minus 119911119894 (119905) minus 11989732 (11)

The amount of the spring compression is

Δ1199092119894 (119905) = 11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732 (12)

The oscillator acceleration yields

119886119894 (119905) = 1198961Δ1199091119894 (119905) + 119898119892 minus 1198962Δ1199092119894 (119905)119898 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732) + 119898119892 minus 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119898 (13)

Then the differential equation of motion for the springoscillator is

1198892119911119894 (119905)1198891199052 + (1198961 + 1198962)119898 119911119894 (119905)= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898 (14)

The above equation is an ordinary differential equationof the second order which can be solved by the Runge-Kuttamethod of the fourth orderThis equation can be transformedto following forms

11991110158401015840119894 (119905) = 119891 (119905 119911119894 (119905) 1199111015840119894 (119905)) 1199050 le 119905 le 119905119899119911119894 (1199050) = 1198971 + 11989732 1199111015840119894 (1199050) = 0

(15)

6 Shock and Vibration

o

d

m

A

ym1

vm1

k1

k2

B

y

l3

l2 = l20 minus Δx21

ln0

l1 = l10 minus Δx11

PA(t)

PB (t)

Figure 7 Initial state of the fixed-length vibration

Assuming that 119908119894(119905) = 1199111015840119894 (119905) the above equations can bewritten as following ordinary differential equations of the firstorder1199111015840119894 (119905) = 119908119894 (119905) 119911119894 (1199050) = 1198971 + 11989732

1199050 le 119905 le 1199051198991199081015840119894 (119905) = 119891 (119905 119911119894 (119905) 119908119894 (119905))= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898minus (1198961 + 1198962)119898 119911119894 (119905)

(16)

According to the Runge-Kuttamethod of the fourth order[17 18] its numerical calculation formula can be representedas

119911119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119911119894119895 (1199050 + 119895ℎ) + ℎ6 (1198701 + 21198702 + 21198703 + 1198704)

119908119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119908119894119895 (1199050 + 119895ℎ) + ℎ6 (1198721 + 21198722 + 21198723 + 1198724)

(17)

where

1198701 = 119908119894119895 (1199050 + 119895ℎ) 1198721 = 11989611198981199061119894 (1199050 + 119895ℎ) + 11989621198981199062119894 (1199050 + 119895ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 119911119894119895 (1199050 + 119895ℎ)

1198702 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987211198722 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198701]

1198703 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987221198723 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198702]

1198704 = 119908119894119895 (1199050 + 119895ℎ) + ℎ11987231198724 = 11989611198981199061119894 (1199050 + (119895 + 1) ℎ) + 11989621198981199062119894 (1199050 + (119895 + 1) ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ1198703]

(18)

Given the displacement boundary conditions of the waveconverter top and bottom (1199061119894(119905) 1199062119894(119905)) the position coor-dinate 119911119894(119905) and velocity V119894(119905) of the oscillator at the time of119905 can be derived Therefore the amount of the compressionΔ1199091119894(119905) Δ1199092119894(119905) can be obtained

432 Differential Equation ofMotion in SlowUnloading StageIn the slow unloading stage the vertical displacements ofthe wave converter top and bottom have few changes overtime so the length of the wave converter can be consideredas a constant 11989710158400 and 11989710158401198990 are the length and clear length ofthe wave converter at the end of the rapid unloading stagerespectively As is shown in Figure 7 the geometry of thewave converter at the end of the rapid unloading stage isdrawn in the solid line while the dashed line shows the static

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

Δui(t)

002 004 006 008 010 012 014 016 018000Dynamic time (s)

Rapid loadingstage Rapid unloading

stageSlow unloading stage

Figure 3 Typical curve of Δ119906119894(119905) with time for the wave converter

Air

Rock

Distribution layer

Underground structure

Monitoring position for displacements

Rock surface at the wave converter top

Distribution layer surface at the wave converter bottom

Initial pressure Pin caused by the wave converter

Figure 4 Schematic of model 1

while the slow unloading stage corresponds to the slowchange period

3 Calculation Process of DynamicResponses for Structures with the NewBlast-Resistant Method

(1) Establish numerical models and acquire the dataneeded for the calculation of the wave converterrsquosdisplacements in the rapid loading stage and rapidunloading stage Model 1 without wave converters forthe finite element analysis is set up shown in Figure 4In model 1 the distribution layer underground struc-ture and rock are established The initial pressurecaused by the wave converter on the distribution layersurface (at the wave converter bottom) is applied asthe lower spring force divided by the cross-sectional

area of the wave converter Then the explosion posi-tion and blasting load should be applied After thatunder the blasting load the finite element analysismethod is used to calculate the vertical displacement-time curves 1199060119894(119905) and 1199062119894(119905) of the rock surface atthe wave converter top and the distribution layer sur-face at the wave converter bottom respectively Themonitoring positions for displacements are suggestedto adopt the tops and bottoms of 9 wave convertersalong the width direction of the structure shown inFigure 4 Then model 2 of the ground without anyconstruction such as the structure distribution layerand wave converters is built up shown in Figure 5The numerical analysis based on model 2 under thesame explosion condition as model 1 is conducted toobtain the vertical rock pressure-time curve1198751198600(119905) forthe corresponding monitoring positions of the rocksurface in model 1 shown in Figure 5

(2) Solve the differential equation of motion for thespring oscillator and obtain the law of motion inthe rapid loading stage and rapid unloading stage Inmodel 1 the vertical displacement-time curve1199060119894(119905) ofthe rock surface at the wave converter top multipliedby a displacement transfer coefficient 119870119894 (0 lt 119870119894 lt1) is considered as the vertical displacement-timecurve 1199061119894(119905) of the wave converter top in the rapidloading stage and rapid unloading stage of the truesituation The true situation refers to the real under-ground structure with an array of wave convertersand a distribution layer The vertical displacement-time curve 1199062119894(119905) of the distribution layer surface atthe wave converter bottom in model 1 is considereddirectly as the vertical displacement-time curve of thewave converter bottom in the rapid loading stage andrapid unloading stage of the true situation In the truesituation the difference between 1199061119894(119905) and 1199062119894(119905) isthe relative displacement Δ119906119894(119905) between the top andbottom of the wave converter The wave converter isin the rapid loading stage before the relative displace-ment reaches the maximum while it is in the rapidunloading stage during the sharp decrease periodafter the relative displacement reaches the maximumTaking the displacements of the wave converter topand bottom as the boundary condition the differen-tial equation ofmotion for the spring oscillator can besolved to obtain the law ofmotion in the rapid loadingstage and rapid unloading stage Then the stress-timecurve of thewave converter top and bottom in above 2stages can be gotten

(3) Solve the differential equation of motion for thespring oscillator and obtain the law of motion inthe slow unloading stage According to step (2) thelength of the wave converter the displacement (orposition coordinate) and velocity of the oscillator atthe end of the rapid unloading step can be gottenThen the differential equation ofmotion for the springoscillator in the slow unloading stage can be solvedIn the slow unloading stage the length of the wave

4 Shock and Vibration

Rock

Rock

RockRock

Monitoring position for the vertical stress

Rock surface at the wave converter top in model 1

Rock

Figure 5 Schematic of model 2

converter can be deemed as a constant In that casethe calculation results of the dynamic response are alittle dangerous whichwould be safer for the structuredesign After that the stress-time curve of the waveconverter top and bottom in the slow unloading stagecan be gotten

(4) Calculate the dynamic response of the undergroundstructure Firstly the initial pressure 119875in caused bythe wave converter on the distribution layer surfacein model 1 should be deleted Then the stress-timecurves of the wave converter top and bottom in threestages are applied on the rock surface and distributionlayer surface instead of the wave converters Afterthat the blasting load is applied and the dynamiccalculation is conducted to get the dynamic responseof the structure

4 Mechanical Characteristics and MotionEvolution Law of Wave Converters

41 Static Constitutive Relation of the Wave Converter 1198970 isthe initial length of the wave converter while 119886 is the shellthickness So the initial clear length of the wave convertercan be written as 1198971198990 = 1198970 minus 2119886 and 1198971198990 is shown in Figure 6In Figure 6 the dashed line is the position under the staticequilibrium and the solid line is the position at any timeunder the dynamic load 1198961 is the stiffness coefficient of theupper spring whose length in free state is 11989710 Δ11990910 is theinitial amount of compression for upper spring 1198962 is thestiffness coefficient of the lower spring and 11989720 is its length infree stateΔ11990920 is the initial amount of compression for lowerspring 119898 and 1198973 are the mass and height of the mass blockrespectively 119875119860(119905) is the vertical rock pressure on the waveconverter top in the true situation while 1198751198600(119905) is the verticalrock pressure at the same location inmodel 2When the wave

d

PA(t)

o A

Au1i(t)

zi(t)

ln0 + u2i(t)

z

B

B

PB(t)

mi(t)ai(t)

k2

k1

l2 = l20 minus Δx20

l1 = l10 minus Δx10

l3ln0

Figure 6 Deformation process of the wave converter

converter is installed with the initial length 1198970 under staticequilibrium the geometric equation is established yielding

1198971198990 = 1198971 + 1198972 + 1198973 = 11989710 + 11989720 + 1198973 minus Δ11990910 minus Δ11990920 (1)

As for the mass block the balance equation is

119898119892 + 1198961Δ11990910 = 1198962Δ11990920 (2)

Combining (1) and (2) the solutions are

Δ11990910 = 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

Δ11990920 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 1198971198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (3)

1198651 and 1198652 are the force of upper and lower springsrespectively 119904 is the cross-sectional area of thewave converter119889 is the cross-sectional length If the weight of the waveconverterrsquos shells is ignored 1198651 can be expressed as

1198651 (119905) = 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) 119904119875119860 (119905) ge 1198961Δ11990910 (4)

1198652 can be expressed as

1198652 (119905) = 119898119892 + 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) + 119898119892 119904119875119860 (119905) ge 1198961Δ11990910 (5)

Shock and Vibration 5

So the static constitutive relation of the wave converteryields

If 119904119875119860 (119905) lt 1198961Δ11990910Δ119897 = 0 (6)

If 119904119875119860 (119905) ge 1198961Δ11990910Δ119897 = (1198961 + 1198962) 119904119875119860 (119905)11989611198962 + [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]1198962 (1198961 + 1198962) (7)

42 Dynamic Response Partitioning of the Wave ConverterThe rapid loading stage rapid unloading stage and slowunloading stage correspond to the rapid compression stagerapid recovery stage and slow recovery stage respectively Inthe slow recovery stage the length of the wave converter canbe deemed as a constantThe demarcation point of stage 1 andstage 2 is that the relative displacement reaches themaximumThe duration time of stage 2 can be determined by the relativedisplacement-time curve When the rock masses above thestructure are in a wide range of elasticity state the rapidunloading stage can not be ignored but if the rockmasses arein a wide range of plastic state the rapid unloading stage canbe ignored

The computing time of stage 3 is advisable for 1 or 2vibration periods Via a large amount of computations it isconcluded that the computing time of stage 3 can be taken as1 vibration period if stage 2 can not be ignored otherwise itcan be taken as 2 vibration periods

43 Differential Equation of Motion for the Spring Oscillator

431 Differential Equation of Motion in Rapid Loading Stageand Rapid Unloading Stage 119873 is the total number of waveconverters along the width direction of the structure and 119894 isthe serial number of the wave converter For wave converter119894 1199060119894(119905) is the vertical displacement of the rock surface at thewave converter top in model 1 and 1199062119894(119905) (119894 = 1 2 119899)is the vertical displacement of the wave converter bottom in

rapid loading stage and rapid unloading stage of the true situ-ation 1199061119894(119905) is the vertical displacement of the wave convertertop in rapid loading stage and rapid unloading stage of thetrue situation which can be expressed as 1199061119894(119905) = 1198701198941199060119894(119905)119870119894is the displacement transfer coefficient of the wave converterΔ119906119894(119905) is the relative displacement between the top and bot-tom of the wave converter which can be written as Δ119906119894(119905) =1199061119894(119905) minus 1199062119894(119905) At the end of the rapid unloading stage is theinitial state of the slow unloading stage which can be deemedas the fixed-length vibration shown in Figure 7 In Figure 7the dashed line is the position under the static equilibrium inthe fixed-length vibration and the solid line is the positionat the end of the rapid unloading stage

As is shown in Figure 6 V119894(119905) 119886119894(119905) and 119911119894(119905) are oscillatorvelocity oscillator acceleration and position coordinate at thetime of 119905 respectively yielding

V119894 (119905) = 119889119911119894 (119905)119889119905 (8)

The length of the upper spring is

1198971119894 (119905) = 119911119894 (119905) minus 1199061119894 (119905) minus 11989732 (9)

The amount of the spring compression is

Δ1199091119894 (119905) = 11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732 (10)

The length of the lower spring is

1198972119894 (119905) = 1198971198990 + 1199062119894 (119905) minus 119911119894 (119905) minus 11989732 (11)

The amount of the spring compression is

Δ1199092119894 (119905) = 11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732 (12)

The oscillator acceleration yields

119886119894 (119905) = 1198961Δ1199091119894 (119905) + 119898119892 minus 1198962Δ1199092119894 (119905)119898 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732) + 119898119892 minus 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119898 (13)

Then the differential equation of motion for the springoscillator is

1198892119911119894 (119905)1198891199052 + (1198961 + 1198962)119898 119911119894 (119905)= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898 (14)

The above equation is an ordinary differential equationof the second order which can be solved by the Runge-Kuttamethod of the fourth orderThis equation can be transformedto following forms

11991110158401015840119894 (119905) = 119891 (119905 119911119894 (119905) 1199111015840119894 (119905)) 1199050 le 119905 le 119905119899119911119894 (1199050) = 1198971 + 11989732 1199111015840119894 (1199050) = 0

(15)

6 Shock and Vibration

o

d

m

A

ym1

vm1

k1

k2

B

y

l3

l2 = l20 minus Δx21

ln0

l1 = l10 minus Δx11

PA(t)

PB (t)

Figure 7 Initial state of the fixed-length vibration

Assuming that 119908119894(119905) = 1199111015840119894 (119905) the above equations can bewritten as following ordinary differential equations of the firstorder1199111015840119894 (119905) = 119908119894 (119905) 119911119894 (1199050) = 1198971 + 11989732

1199050 le 119905 le 1199051198991199081015840119894 (119905) = 119891 (119905 119911119894 (119905) 119908119894 (119905))= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898minus (1198961 + 1198962)119898 119911119894 (119905)

(16)

According to the Runge-Kuttamethod of the fourth order[17 18] its numerical calculation formula can be representedas

119911119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119911119894119895 (1199050 + 119895ℎ) + ℎ6 (1198701 + 21198702 + 21198703 + 1198704)

119908119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119908119894119895 (1199050 + 119895ℎ) + ℎ6 (1198721 + 21198722 + 21198723 + 1198724)

(17)

where

1198701 = 119908119894119895 (1199050 + 119895ℎ) 1198721 = 11989611198981199061119894 (1199050 + 119895ℎ) + 11989621198981199062119894 (1199050 + 119895ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 119911119894119895 (1199050 + 119895ℎ)

1198702 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987211198722 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198701]

1198703 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987221198723 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198702]

1198704 = 119908119894119895 (1199050 + 119895ℎ) + ℎ11987231198724 = 11989611198981199061119894 (1199050 + (119895 + 1) ℎ) + 11989621198981199062119894 (1199050 + (119895 + 1) ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ1198703]

(18)

Given the displacement boundary conditions of the waveconverter top and bottom (1199061119894(119905) 1199062119894(119905)) the position coor-dinate 119911119894(119905) and velocity V119894(119905) of the oscillator at the time of119905 can be derived Therefore the amount of the compressionΔ1199091119894(119905) Δ1199092119894(119905) can be obtained

432 Differential Equation ofMotion in SlowUnloading StageIn the slow unloading stage the vertical displacements ofthe wave converter top and bottom have few changes overtime so the length of the wave converter can be consideredas a constant 11989710158400 and 11989710158401198990 are the length and clear length ofthe wave converter at the end of the rapid unloading stagerespectively As is shown in Figure 7 the geometry of thewave converter at the end of the rapid unloading stage isdrawn in the solid line while the dashed line shows the static

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

Rock

Rock

RockRock

Monitoring position for the vertical stress

Rock surface at the wave converter top in model 1

Rock

Figure 5 Schematic of model 2

converter can be deemed as a constant In that casethe calculation results of the dynamic response are alittle dangerous whichwould be safer for the structuredesign After that the stress-time curve of the waveconverter top and bottom in the slow unloading stagecan be gotten

(4) Calculate the dynamic response of the undergroundstructure Firstly the initial pressure 119875in caused bythe wave converter on the distribution layer surfacein model 1 should be deleted Then the stress-timecurves of the wave converter top and bottom in threestages are applied on the rock surface and distributionlayer surface instead of the wave converters Afterthat the blasting load is applied and the dynamiccalculation is conducted to get the dynamic responseof the structure

4 Mechanical Characteristics and MotionEvolution Law of Wave Converters

41 Static Constitutive Relation of the Wave Converter 1198970 isthe initial length of the wave converter while 119886 is the shellthickness So the initial clear length of the wave convertercan be written as 1198971198990 = 1198970 minus 2119886 and 1198971198990 is shown in Figure 6In Figure 6 the dashed line is the position under the staticequilibrium and the solid line is the position at any timeunder the dynamic load 1198961 is the stiffness coefficient of theupper spring whose length in free state is 11989710 Δ11990910 is theinitial amount of compression for upper spring 1198962 is thestiffness coefficient of the lower spring and 11989720 is its length infree stateΔ11990920 is the initial amount of compression for lowerspring 119898 and 1198973 are the mass and height of the mass blockrespectively 119875119860(119905) is the vertical rock pressure on the waveconverter top in the true situation while 1198751198600(119905) is the verticalrock pressure at the same location inmodel 2When the wave

d

PA(t)

o A

Au1i(t)

zi(t)

ln0 + u2i(t)

z

B

B

PB(t)

mi(t)ai(t)

k2

k1

l2 = l20 minus Δx20

l1 = l10 minus Δx10

l3ln0

Figure 6 Deformation process of the wave converter

converter is installed with the initial length 1198970 under staticequilibrium the geometric equation is established yielding

1198971198990 = 1198971 + 1198972 + 1198973 = 11989710 + 11989720 + 1198973 minus Δ11990910 minus Δ11990920 (1)

As for the mass block the balance equation is

119898119892 + 1198961Δ11990910 = 1198962Δ11990920 (2)

Combining (1) and (2) the solutions are

Δ11990910 = 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

Δ11990920 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 1198971198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (3)

1198651 and 1198652 are the force of upper and lower springsrespectively 119904 is the cross-sectional area of thewave converter119889 is the cross-sectional length If the weight of the waveconverterrsquos shells is ignored 1198651 can be expressed as

1198651 (119905) = 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) 119904119875119860 (119905) ge 1198961Δ11990910 (4)

1198652 can be expressed as

1198652 (119905) = 119898119892 + 1198961Δ11990910 119904119875119860 (119905) lt 1198961Δ11990910119904119875119860 (119905) + 119898119892 119904119875119860 (119905) ge 1198961Δ11990910 (5)

Shock and Vibration 5

So the static constitutive relation of the wave converteryields

If 119904119875119860 (119905) lt 1198961Δ11990910Δ119897 = 0 (6)

If 119904119875119860 (119905) ge 1198961Δ11990910Δ119897 = (1198961 + 1198962) 119904119875119860 (119905)11989611198962 + [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]1198962 (1198961 + 1198962) (7)

42 Dynamic Response Partitioning of the Wave ConverterThe rapid loading stage rapid unloading stage and slowunloading stage correspond to the rapid compression stagerapid recovery stage and slow recovery stage respectively Inthe slow recovery stage the length of the wave converter canbe deemed as a constantThe demarcation point of stage 1 andstage 2 is that the relative displacement reaches themaximumThe duration time of stage 2 can be determined by the relativedisplacement-time curve When the rock masses above thestructure are in a wide range of elasticity state the rapidunloading stage can not be ignored but if the rockmasses arein a wide range of plastic state the rapid unloading stage canbe ignored

The computing time of stage 3 is advisable for 1 or 2vibration periods Via a large amount of computations it isconcluded that the computing time of stage 3 can be taken as1 vibration period if stage 2 can not be ignored otherwise itcan be taken as 2 vibration periods

43 Differential Equation of Motion for the Spring Oscillator

431 Differential Equation of Motion in Rapid Loading Stageand Rapid Unloading Stage 119873 is the total number of waveconverters along the width direction of the structure and 119894 isthe serial number of the wave converter For wave converter119894 1199060119894(119905) is the vertical displacement of the rock surface at thewave converter top in model 1 and 1199062119894(119905) (119894 = 1 2 119899)is the vertical displacement of the wave converter bottom in

rapid loading stage and rapid unloading stage of the true situ-ation 1199061119894(119905) is the vertical displacement of the wave convertertop in rapid loading stage and rapid unloading stage of thetrue situation which can be expressed as 1199061119894(119905) = 1198701198941199060119894(119905)119870119894is the displacement transfer coefficient of the wave converterΔ119906119894(119905) is the relative displacement between the top and bot-tom of the wave converter which can be written as Δ119906119894(119905) =1199061119894(119905) minus 1199062119894(119905) At the end of the rapid unloading stage is theinitial state of the slow unloading stage which can be deemedas the fixed-length vibration shown in Figure 7 In Figure 7the dashed line is the position under the static equilibrium inthe fixed-length vibration and the solid line is the positionat the end of the rapid unloading stage

As is shown in Figure 6 V119894(119905) 119886119894(119905) and 119911119894(119905) are oscillatorvelocity oscillator acceleration and position coordinate at thetime of 119905 respectively yielding

V119894 (119905) = 119889119911119894 (119905)119889119905 (8)

The length of the upper spring is

1198971119894 (119905) = 119911119894 (119905) minus 1199061119894 (119905) minus 11989732 (9)

The amount of the spring compression is

Δ1199091119894 (119905) = 11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732 (10)

The length of the lower spring is

1198972119894 (119905) = 1198971198990 + 1199062119894 (119905) minus 119911119894 (119905) minus 11989732 (11)

The amount of the spring compression is

Δ1199092119894 (119905) = 11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732 (12)

The oscillator acceleration yields

119886119894 (119905) = 1198961Δ1199091119894 (119905) + 119898119892 minus 1198962Δ1199092119894 (119905)119898 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732) + 119898119892 minus 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119898 (13)

Then the differential equation of motion for the springoscillator is

1198892119911119894 (119905)1198891199052 + (1198961 + 1198962)119898 119911119894 (119905)= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898 (14)

The above equation is an ordinary differential equationof the second order which can be solved by the Runge-Kuttamethod of the fourth orderThis equation can be transformedto following forms

11991110158401015840119894 (119905) = 119891 (119905 119911119894 (119905) 1199111015840119894 (119905)) 1199050 le 119905 le 119905119899119911119894 (1199050) = 1198971 + 11989732 1199111015840119894 (1199050) = 0

(15)

6 Shock and Vibration

o

d

m

A

ym1

vm1

k1

k2

B

y

l3

l2 = l20 minus Δx21

ln0

l1 = l10 minus Δx11

PA(t)

PB (t)

Figure 7 Initial state of the fixed-length vibration

Assuming that 119908119894(119905) = 1199111015840119894 (119905) the above equations can bewritten as following ordinary differential equations of the firstorder1199111015840119894 (119905) = 119908119894 (119905) 119911119894 (1199050) = 1198971 + 11989732

1199050 le 119905 le 1199051198991199081015840119894 (119905) = 119891 (119905 119911119894 (119905) 119908119894 (119905))= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898minus (1198961 + 1198962)119898 119911119894 (119905)

(16)

According to the Runge-Kuttamethod of the fourth order[17 18] its numerical calculation formula can be representedas

119911119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119911119894119895 (1199050 + 119895ℎ) + ℎ6 (1198701 + 21198702 + 21198703 + 1198704)

119908119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119908119894119895 (1199050 + 119895ℎ) + ℎ6 (1198721 + 21198722 + 21198723 + 1198724)

(17)

where

1198701 = 119908119894119895 (1199050 + 119895ℎ) 1198721 = 11989611198981199061119894 (1199050 + 119895ℎ) + 11989621198981199062119894 (1199050 + 119895ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 119911119894119895 (1199050 + 119895ℎ)

1198702 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987211198722 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198701]

1198703 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987221198723 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198702]

1198704 = 119908119894119895 (1199050 + 119895ℎ) + ℎ11987231198724 = 11989611198981199061119894 (1199050 + (119895 + 1) ℎ) + 11989621198981199062119894 (1199050 + (119895 + 1) ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ1198703]

(18)

Given the displacement boundary conditions of the waveconverter top and bottom (1199061119894(119905) 1199062119894(119905)) the position coor-dinate 119911119894(119905) and velocity V119894(119905) of the oscillator at the time of119905 can be derived Therefore the amount of the compressionΔ1199091119894(119905) Δ1199092119894(119905) can be obtained

432 Differential Equation ofMotion in SlowUnloading StageIn the slow unloading stage the vertical displacements ofthe wave converter top and bottom have few changes overtime so the length of the wave converter can be consideredas a constant 11989710158400 and 11989710158401198990 are the length and clear length ofthe wave converter at the end of the rapid unloading stagerespectively As is shown in Figure 7 the geometry of thewave converter at the end of the rapid unloading stage isdrawn in the solid line while the dashed line shows the static

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

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Active and Passive Electronic Components

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

So the static constitutive relation of the wave converteryields

If 119904119875119860 (119905) lt 1198961Δ11990910Δ119897 = 0 (6)

If 119904119875119860 (119905) ge 1198961Δ11990910Δ119897 = (1198961 + 1198962) 119904119875119860 (119905)11989611198962 + [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]1198962 (1198961 + 1198962) (7)

42 Dynamic Response Partitioning of the Wave ConverterThe rapid loading stage rapid unloading stage and slowunloading stage correspond to the rapid compression stagerapid recovery stage and slow recovery stage respectively Inthe slow recovery stage the length of the wave converter canbe deemed as a constantThe demarcation point of stage 1 andstage 2 is that the relative displacement reaches themaximumThe duration time of stage 2 can be determined by the relativedisplacement-time curve When the rock masses above thestructure are in a wide range of elasticity state the rapidunloading stage can not be ignored but if the rockmasses arein a wide range of plastic state the rapid unloading stage canbe ignored

The computing time of stage 3 is advisable for 1 or 2vibration periods Via a large amount of computations it isconcluded that the computing time of stage 3 can be taken as1 vibration period if stage 2 can not be ignored otherwise itcan be taken as 2 vibration periods

43 Differential Equation of Motion for the Spring Oscillator

431 Differential Equation of Motion in Rapid Loading Stageand Rapid Unloading Stage 119873 is the total number of waveconverters along the width direction of the structure and 119894 isthe serial number of the wave converter For wave converter119894 1199060119894(119905) is the vertical displacement of the rock surface at thewave converter top in model 1 and 1199062119894(119905) (119894 = 1 2 119899)is the vertical displacement of the wave converter bottom in

rapid loading stage and rapid unloading stage of the true situ-ation 1199061119894(119905) is the vertical displacement of the wave convertertop in rapid loading stage and rapid unloading stage of thetrue situation which can be expressed as 1199061119894(119905) = 1198701198941199060119894(119905)119870119894is the displacement transfer coefficient of the wave converterΔ119906119894(119905) is the relative displacement between the top and bot-tom of the wave converter which can be written as Δ119906119894(119905) =1199061119894(119905) minus 1199062119894(119905) At the end of the rapid unloading stage is theinitial state of the slow unloading stage which can be deemedas the fixed-length vibration shown in Figure 7 In Figure 7the dashed line is the position under the static equilibrium inthe fixed-length vibration and the solid line is the positionat the end of the rapid unloading stage

As is shown in Figure 6 V119894(119905) 119886119894(119905) and 119911119894(119905) are oscillatorvelocity oscillator acceleration and position coordinate at thetime of 119905 respectively yielding

V119894 (119905) = 119889119911119894 (119905)119889119905 (8)

The length of the upper spring is

1198971119894 (119905) = 119911119894 (119905) minus 1199061119894 (119905) minus 11989732 (9)

The amount of the spring compression is

Δ1199091119894 (119905) = 11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732 (10)

The length of the lower spring is

1198972119894 (119905) = 1198971198990 + 1199062119894 (119905) minus 119911119894 (119905) minus 11989732 (11)

The amount of the spring compression is

Δ1199092119894 (119905) = 11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732 (12)

The oscillator acceleration yields

119886119894 (119905) = 1198961Δ1199091119894 (119905) + 119898119892 minus 1198962Δ1199092119894 (119905)119898 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732) + 119898119892 minus 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119898 (13)

Then the differential equation of motion for the springoscillator is

1198892119911119894 (119905)1198891199052 + (1198961 + 1198962)119898 119911119894 (119905)= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898 (14)

The above equation is an ordinary differential equationof the second order which can be solved by the Runge-Kuttamethod of the fourth orderThis equation can be transformedto following forms

11991110158401015840119894 (119905) = 119891 (119905 119911119894 (119905) 1199111015840119894 (119905)) 1199050 le 119905 le 119905119899119911119894 (1199050) = 1198971 + 11989732 1199111015840119894 (1199050) = 0

(15)

6 Shock and Vibration

o

d

m

A

ym1

vm1

k1

k2

B

y

l3

l2 = l20 minus Δx21

ln0

l1 = l10 minus Δx11

PA(t)

PB (t)

Figure 7 Initial state of the fixed-length vibration

Assuming that 119908119894(119905) = 1199111015840119894 (119905) the above equations can bewritten as following ordinary differential equations of the firstorder1199111015840119894 (119905) = 119908119894 (119905) 119911119894 (1199050) = 1198971 + 11989732

1199050 le 119905 le 1199051198991199081015840119894 (119905) = 119891 (119905 119911119894 (119905) 119908119894 (119905))= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898minus (1198961 + 1198962)119898 119911119894 (119905)

(16)

According to the Runge-Kuttamethod of the fourth order[17 18] its numerical calculation formula can be representedas

119911119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119911119894119895 (1199050 + 119895ℎ) + ℎ6 (1198701 + 21198702 + 21198703 + 1198704)

119908119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119908119894119895 (1199050 + 119895ℎ) + ℎ6 (1198721 + 21198722 + 21198723 + 1198724)

(17)

where

1198701 = 119908119894119895 (1199050 + 119895ℎ) 1198721 = 11989611198981199061119894 (1199050 + 119895ℎ) + 11989621198981199062119894 (1199050 + 119895ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 119911119894119895 (1199050 + 119895ℎ)

1198702 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987211198722 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198701]

1198703 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987221198723 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198702]

1198704 = 119908119894119895 (1199050 + 119895ℎ) + ℎ11987231198724 = 11989611198981199061119894 (1199050 + (119895 + 1) ℎ) + 11989621198981199062119894 (1199050 + (119895 + 1) ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ1198703]

(18)

Given the displacement boundary conditions of the waveconverter top and bottom (1199061119894(119905) 1199062119894(119905)) the position coor-dinate 119911119894(119905) and velocity V119894(119905) of the oscillator at the time of119905 can be derived Therefore the amount of the compressionΔ1199091119894(119905) Δ1199092119894(119905) can be obtained

432 Differential Equation ofMotion in SlowUnloading StageIn the slow unloading stage the vertical displacements ofthe wave converter top and bottom have few changes overtime so the length of the wave converter can be consideredas a constant 11989710158400 and 11989710158401198990 are the length and clear length ofthe wave converter at the end of the rapid unloading stagerespectively As is shown in Figure 7 the geometry of thewave converter at the end of the rapid unloading stage isdrawn in the solid line while the dashed line shows the static

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

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RotatingMachinery

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

o

d

m

A

ym1

vm1

k1

k2

B

y

l3

l2 = l20 minus Δx21

ln0

l1 = l10 minus Δx11

PA(t)

PB (t)

Figure 7 Initial state of the fixed-length vibration

Assuming that 119908119894(119905) = 1199111015840119894 (119905) the above equations can bewritten as following ordinary differential equations of the firstorder1199111015840119894 (119905) = 119908119894 (119905) 119911119894 (1199050) = 1198971 + 11989732

1199050 le 119905 le 1199051198991199081015840119894 (119905) = 119891 (119905 119911119894 (119905) 119908119894 (119905))= 11989611198981199061119894 (119905) + 11989621198981199062119894 (119905)

minus 1198962 (11989720 minus 1198971198990) minus 119898119892 minus 119896111989710 minus (1198961 minus 1198962) 11989732119898minus (1198961 + 1198962)119898 119911119894 (119905)

(16)

According to the Runge-Kuttamethod of the fourth order[17 18] its numerical calculation formula can be representedas

119911119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119911119894119895 (1199050 + 119895ℎ) + ℎ6 (1198701 + 21198702 + 21198703 + 1198704)

119908119894(119895+1) (1199050 + (119895 + 1) ℎ)= 119908119894119895 (1199050 + 119895ℎ) + ℎ6 (1198721 + 21198722 + 21198723 + 1198724)

(17)

where

1198701 = 119908119894119895 (1199050 + 119895ℎ) 1198721 = 11989611198981199061119894 (1199050 + 119895ℎ) + 11989621198981199062119894 (1199050 + 119895ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 119911119894119895 (1199050 + 119895ℎ)

1198702 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987211198722 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198701]

1198703 = 119908119894119895 (1199050 + 119895ℎ) + ℎ211987221198723 = 11989611198981199061119894 (1199050 + 119895ℎ + ℎ2) + 11989621198981199062119894 (1199050 + 119895ℎ + ℎ2)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ21198702]

1198704 = 119908119894119895 (1199050 + 119895ℎ) + ℎ11987231198724 = 11989611198981199061119894 (1199050 + (119895 + 1) ℎ) + 11989621198981199062119894 (1199050 + (119895 + 1) ℎ)

minus 1198962 (11989720 minus 1198970) minus 119898119892 minus 119896111989710119898minus (1198961 + 1198962)119898 [119911119894119895 (1199050 + 119895ℎ) + ℎ1198703]

(18)

Given the displacement boundary conditions of the waveconverter top and bottom (1199061119894(119905) 1199062119894(119905)) the position coor-dinate 119911119894(119905) and velocity V119894(119905) of the oscillator at the time of119905 can be derived Therefore the amount of the compressionΔ1199091119894(119905) Δ1199092119894(119905) can be obtained

432 Differential Equation ofMotion in SlowUnloading StageIn the slow unloading stage the vertical displacements ofthe wave converter top and bottom have few changes overtime so the length of the wave converter can be consideredas a constant 11989710158400 and 11989710158401198990 are the length and clear length ofthe wave converter at the end of the rapid unloading stagerespectively As is shown in Figure 7 the geometry of thewave converter at the end of the rapid unloading stage isdrawn in the solid line while the dashed line shows the static

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

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Shock and Vibration

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International Journal of

Shock and Vibration 7

equilibrium position of the mass block The 119910-axis is verticaldownward and its coordinate origin is the static equilibriumposition of the mass block in stage 3 In Figure 7 at the endof the rapid unloading stage V1198981 is the oscillator velocity and1199101198981 is the distance from the oscillator center to that of thestatic equilibrium position

If the oscillator is in static equilibrium under the con-verter length of 11989710158400 11989710158401198990 yields

11989710158401198990 = 11989710 + 11989720 + 1198973 minus Δ11990911 minus Δ11990921 (19)

The balance equation for the mass block is

119898119892 + 1198961Δ11990911 = 1198962Δ11990921 (20)

where Δ11990911 and Δ11990921 are the amount of compression forupper and lower spring in static equilibrium under theconverter length of 11989710158400 respectively Δ11990911 and Δ11990921 are

Δ11990911 = 1198962 (11989710 + 11989720 + 1198973 minus 11989710158401198990) minus 1198981198921198961 + 1198962

Δ11990921 = 1198981198921198962 + 1198961 (11989710 + 11989720 + 1198973 minus 11989710158401198990)1198961 + 1198962 minus 11989611198981198921198962 (1198961 + 1198962) (21)

119865119894 119891119868119894 and 119878119894 are the active force inertia force and con-straint counterforce respectively Based on the DrsquoAlembertprinciple 119865119894 119891119868119894 and 119878119894 yield

119865119894 + 119878119894 + 119891119868119894 = 0 (22)

The active force consists of the gravity force119898119892 dampingforce 119891119863 and elastic restoring force 119891119904 Then (22) can bewritten as

119898 119910119898 (119905) + 119888 119910119898 (119905) + (1198961 + 1198962) 119910119898 (119905) = 0 (23)

119910119898(119905) is the position coordinate of the oscillator in 119910 coor-dinate The time at the end of the rapid unloading stage isassumed as 119905119906 and then the position coordinate and velocityof the oscillator can be expressed as 119911119894(119905119906) and V119894(119905119906) in119911 coordinate Via the coordinate transform the positioncoordinate and velocity of the oscillator in 119910 coordinate canbe expressed as 1199101198981 and V1198981 1199101198981 and V1198981 are the initialconditions of (23) so the solution of (23) is

119910119898 (119905) = 119860 cos120596119863119905 + 119861 sin120596119863119905 (24)

where

119860 = 1199101198981119890minus120577120596119899119905119861 = V1198981 + 1205771205961198991199101198981120596119863 119890minus120577120596119899119905

120596119863 = 120596119899radic1 minus 1205772120596119899 = radic1198961 + 1198962119898

(25)

120577 is the damping ratio If 120577 = 0 the calculation results of thedynamic response are a little dangerous which would be saferfor the structure design so 120577 is considered as 0 in the analysisof stage 3 Considering that 119910119898(119905) is solved the amount ofcompression for springs at any time can be obtained Afterthat the stress-time curve of the wave converter top andbottom in the slow unloading stage can be gotten

44 Displacement Transfer Coefficient of the Wave ConverterIn the rapid loading stage and rapid unloading stage basedon the numerical calculations in model 1 and model 2 thefollowing equation can be derived

1199041198751198600 (119905) 1199081199060119894 (119905) = 1199041198751198600 (119905) 119908 minus 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)]1198701198941199060119894 (119905) (26)

where 119908 is the area ratio of the wave converterrsquos cross-sectional area and 119908 is equal to all wave convertersrsquo cross-sectional areas divided by the area of rock surface abovethe structure According to (4) and (7) when 119904119875119860(119905) ge1198961Δ11990910 the constitutive relation for the wave converter canbe modified as

1198701198941199060119894 (119905) minus 1199062119894 (119905)= 1198651 [1198701198941199060119894 (119905) minus 1199062119894 (119905)] 1198961 + 119896211989611198962

minus (1198961 + 1198962) (11989710 + 11989720 + 1198973 minus 1198971198990) minus 1198981198921198961 + 1198962

+ 11989611198981198921198962 (1198961 + 1198962)

(27)

Combining (26) and (27) 119870119894 yields

119870119894 = (1198961 + 1198962) 1199041198751198600 (119905) + 119908119896111989621199062119894 (119905) + 1199081198961 [119898119892 minus 1198962 (11989710 + 11989720 + 1198973 minus 1198971198990)]119908119896111989621199060119894 (119905) + (1198961 + 1198962) 1199041198751198600 (119905) (28)

45 Stress Inversion of the Wave Converter Based on thesolutions on differential equations ofmotion in 3 stages if thegravity force of the wave converter shell is ignored the stress-time curves of the wave converter top and bottom in 3 stagescan be gotten

In stage 1 and stage 2 the function of the stress-time curveof the wave converter top is

1198751119894 (119905) = 1198961Δ1199091119894 (119905)119904 = 1198961 (11989710 minus 119911119894 (119905) + 1199061119894 (119905) + 11989732)119904 (29)

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

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RotatingMachinery

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Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Navigation and Observation

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

Table 1 Mechanical parameters for the rock

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)2400 13 028 11 45 091

Wav

eco

nver

ter 1

Wav

eco

nver

ter 2

Wav

eco

nver

ter 3

Wav

eco

nver

ter 4

Wav

eco

nver

ter 5

Sect

ion

5

Sect

ion

4

Sect

ion

3

Sect

ion

2

Sect

ion

1

Section 9

Section 10

Section 8

Section 7

Section 6

Sect

ion

11

Sect

ion

12

Sect

ion

13

Sect

ion

14

Sect

ion

15

02

m22

m005

m0

8m

ℎn=

06

m

06 m 06 mln = 4 m

y

O Xln8 ln8 ln8 ln8

ℎn8

ℎn8

ℎn8

ℎn8

Figure 8 Structure size and monitoring sections for internal forces in case 1

The function of the stress-time curve of the wave con-verter bottom is

1198752119894 (119905) = 1198962Δ1199092119894 (119905)119904= 1198962 (11989720 minus 1198971198990 minus 1199062119894 (119905) + 119911119894 (119905) + 11989732)119904

(30)

In stage 3 the functions of the stress-time curves of thewave converter top and bottom are respectively

1198751119894 (119905) = 1198961 (Δ11990911 minus 119910119898 (119905))119904= 1198961Δ11990911 minus 1198961 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

1198752119894 (119905) = 1198962 (Δ11990921 + 119910119898 (119905))119904= 1198962Δ11990921 + 1198962 (119860 cos120596119899119905 + 119861 sin120596119899119905)119904

(31)

Then in model 1 the initial pressure 119875in caused by the waveconverter on the distribution layer surface is deleted and thestress-time curves of the wave converter top and bottom inthree stages are applied on the rock surface and distributionlayer surface respectively After that the blasting load isapplied and the dynamic calculation is conducted to getthe dynamic response of the structure (shown in calculationprocess (4))

5 Case Study

51 Case Set-Up Based on the software of FLAC3D 2 kindsof cases are set up to conduct the dynamic analysis Case 1 isthe traditional underground structure without the new blast-resistant method while case 2 is the underground structurewith the mentioned new blast-resistant method Comparedwith case 2 the difference in case 1 is that the wave converterand distribution layer are not set

Figure 8 shows the structure size andmonitoring sectionsfor internal forces in case 2 and the monitoring sections forinternal forces are the same as case 1The size of the structureis designed according to literature [19] and the buried depthis 10m Mechanical parameters for the rock and structure arelisted in Tables 1 and 2 The distribution layer is made up ofthe foam concrete which has the density of 799 kgm3 andthickness of 005m The mechanical parameters of the foamconcrete are shown inTable 3 [20] In case 2 the tops and bot-toms of wave converters 1sim5 are chosen as monitoring posi-tions for displacements and stresses shown in Figure 8 andthe interpolation is used to get the stresses of other wave con-verters The wave converter is a cubic structure with the sidelength of 02m 1198970 = 02m 119886 = 005m 119908 = 100 1198961 = 1198962 =100 kNm 11989710 = 11989720 = 005m 1198973 = 014m119898 = 389 kg

The width height and thickness of the numerical modelsare 352m 2885m and 1m respectively The blasting loadis assumed as a triangle wave acting on the ground surface(in Figure 9) and the loading scope is from minus3m to 3m on119909-axis The lifting duration and drop duration of the blastpressure-time curve are set as 1ms and 6ms according tothe literature [21] respectively The peak of the shock wave

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

Table 2 Mechanical parameters for the structure

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘)2400 30 021 368 587

Table 3 Mechanical parameters for foam concrete

Density (kgm3) Elastic modulus (GPa) Poisson ratio Cohesion (MPa) Internal friction angle (∘) Tensile strength (MPa)799 0342 01 017 29 02

p

pm

o t1 t2

t

Figure 9 Curve of the blasting load

Figure 10 Model 1 built up based on FLAC3D for numericalcalculation

pressure 119901119898 is 06GPa Model 1 built up based on FLAC3Dfor numerical calculation is shown in Figure 10

According to the symmetry the monitoring positions fordisplacements are suggested to adopt the tops and bottoms of5 wave converters along the width direction of the structureshown in Figure 8 In order to obtain the internal forces suchas the bending moment axial force and shear force of thestructure a total of 15 monitoring sections are set up shownin Figure 8

52 Analysis of the Calculation Results The curves of thedisplacement transfer coefficient 119870119894 with time in the rapid

075

080

085

090

095

100

Disp

lace

men

t tra

nsfe

r coe

ffici

entK

i

002 004 006 008 010 012000Dynamic time (s)

K1

K2

K3

K4

K5

Figure 11 Curves of the displacement transfer coefficient with time

loading and rapid unloading stages are shown in Figure 11and 119894 represents the serial number of the wave converterThe curves of 1199060119894(119905) 1199061119894(119905) 1199062119894(119905) and Δ119906119894(119905) with time areshown in Figures 12ndash15 respectively In all 3 stages of thewaveconverter deformation the stress-time curves of the waveconverter top are shown in Figure 16 while the stress-timecurves of the wave converter bottom are shown in Figure 17The internal forces such as the bending moment axial forceand shear force are listed in Tables 4ndash6

In Tables 4ndash6 the bending moment resulting in the ten-sile stress in the inner element of the structure is positiveThepositive shear force is by counterclockwise while the axialforce to tension is positive It can be concluded that the peakabsolute value of the bending moment in case 2 is generallylower than that of case 1 and the maximum drop in the roofside wall and floor is 573 697 and 527 respectively

The peak absolute value of the shear force for monitoringsections in case 2 is also generally lower than that of case 1The maximum drop in the roof side wall and floor is 862756 and 331 respectivelyThe peak absolute value of theaxial force for monitoring sections in case 2 is remarkablylower than that of case 1 The maximum drop in the roofside wall and floor is 447 747 and 372 respectivelyFor the roof the decrease of the axial tensile force near themidspan is obvious and the maximum drop occurs to the

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

00000

00005

00010

00015

00020

002 004 006 008 010 012000Dynamic time (s)

u0i(t)

(m)

Figure 12 Curves of 1199060119894(119905) with time

minus00002

0000000002000040000600008000100001200014000160001800020

u1i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 13 Curves of 1199061119894(119905) with time

span center The decrease of the axial tensile stress presentsthat the tensile failure in the roof can be alleviated via thewave converters

The curves of the vertical normal stresses for inner andouter elements in the span center with time are shownin Figure 18 The positive value stands for the verticaltensile stress while the negative value stands for the verticalcompressive stress After the adoption of wave convertersthe peak vertical tensile stress for inner element in the spancenter drops from 0055MPa to 0019MPa whichmeans thatthe possibility of spalling damage for roof is reduced Thepeak vertical compressive stress for outer element in the spancenter drops from 173MPa to 0153MPa which means thatthe impact load acting on the roof is reduced

The peak horizontal tensile stresses of monitoring sec-tions for roof in 2 cases are shown in Figure 19 For case 2

minus000005

000000

000005

000010

000015

000020

000025

000030

000035

000040

000045

u2i(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 14 Curves of 1199062119894(119905) with time

minus00002

00000000020000400006000080001000012000140001600018

Δui(t)

(m)

002 004 006 008 010 012000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 15 Curves of Δ119906119894(119905) with time

with wave converters the peak horizontal tensile stressesof inner elements for roof are generally lower than that ofcase 1 with a maximum decrease of 60 Though the peakhorizontal tensile stresses of outer elements for roof becomehigher than that of case 1 the peak horizontal tensile stressesafter increasing are not very large on thewhole Via increasingthe spring stiffness the increase of the peak horizontal tensilestresses of outer elements for roof can be adjusted In thedesign process of wave converters the vibration period of thespring oscillator should not be next to the vibration period ofthe underground structure

6 Conclusions

In this paper a new blast-resistant method based on waveconverters with spring oscillator for underground structuresis put forwardThe conception and calculation process of this

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

Table 4 Internal forces of the structure for monitoring sections 1ndash5

Monitoring section 1 2 3 4 5Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 733 minus43847 640 minus45141 365 minus56001 243 minus92795 minus290760 minus129805Maximum 221068 163402 224395 148737 222567 95049 118881 8908 348 2039

Shear force (N)Minimum minus27317 minus15193 2360 minus3988 1194 minus1956 1177 0 3474 23Maximum minus426 1037 164052 64656 439202 138742 1041408 214362 2040240 282142

Axial force (N)Minimum minus41094 minus83637 minus42568 minus81050 minus57842 minus81420 minus104766 minus83599 minus348627 minus91759Maximum 1666072 921440 1596560 921280 1320330 921600 788366 932400 409633 949376

Table 5 Internal forces of the structure for monitoring sections 6ndash10

Monitoring section 6 7 8 9 10Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum minus246269 minus74504 minus60566 minus34165 minus20880 minus16684 minus20016 minus12038 minus30226 minus27187Maximum minus62 59448 11794 32440 20808 19901 4810 3787 10929 8644

Shear force (N)Minimum minus837708 minus99838 minus371508 minus85756 minus165964 minus58982 minus78672 minus57528 minus100307 minus77861Maximum minus1537 211056 minus1517 90487 minus953 68296 27172 105168 51451 117882

Axial force (N)Minimum minus3218640 minus814500 minus2655720 minus846588 minus2229240 minus891240 minus1992840 minus936840 minus1573560 minus834600Maximum minus4890 minus1196 minus1170 minus2138 minus1175 minus1223 minus620 minus1812 minus5041 minus983

56000

57000

58000

59000

60000

61000

62000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r top

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 16 Stress-time curves of the wave converter top

new method are introduced The mechanical characteristicsand motion evolution law of the wave converter are derivedThe dynamic responses of the traditional underground struc-ture and the new blast-resistant one are also calculatedto verify the blast-resistant effect of the new method Thefollowing conclusions can be drawn through the study

65000

66000

67000

68000

69000

70000

71000

72000

Stre

ss-ti

me c

urve

s of t

he w

ave

conv

erte

r bot

tom

(Pa)

005 010 015 020 025 030000Dynamic time (s)

Wave converter 1Wave converter 2Wave converter 3

Wave converter 4Wave converter 5

Figure 17 Stress-time curves of the wave converter bottom

(1) After the deployment of wave converters the peakabsolute values of the bending moment shear forceand axial force decrease generallyThe decrease of thepeak internal forces means that smaller size and lesssteel are needed in the design of the structure whichcould help reduce the costs

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

Table 6 Internal forces of the structure for monitoring sections 11ndash15

Monitoring section 11 12 13 14 15Case 1 2 1 2 1 2 1 2 1 2Bending moment (N sdotm)

Minimum 230 664 261 614 295 555 97 419 minus54210 minus45028Maximum 35258 23027 36594 23699 39747 24339 32743 15492 15725 14552

Shear force (N)Minimum minus421 minus92 minus19776 minus16431 minus65196 minus51572 minus212922 minus153744 minus533254 minus356511Maximum 4128 3608 319 411 minus428 450 2836 718 5536 6486

Axial force (N)Minimum minus3124 minus4749 minus3185 minus5153 minus2953 minus18050 minus32879 minus115840 minus192165 minus250929Maximum 680160 426831 667594 419751 615300 394217 518211 354017 381111 332100

Case 1Case 2

minus200000

minus150000

minus100000

minus50000

0

50000

Ver

tical

nor

mal

stre

ss o

f inn

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

(a) Inner element

000 001 002 003 004 005

minus200

minus100

minus1800000

minus1600000

minus1400000

minus1200000

minus1000000

minus800000

minus600000

minus400000

minus200000

0

200000

Ver

tical

nor

mal

stre

ss o

f out

er el

emen

t (Pa

)

005 010 015 020 025 030000Dynamic time (s)

Case 1Case 2

times103

(b) Outer element

Figure 18 Vertical normal stress of inner and outer elements in the span center

Inner element of case 1Inner element of case 2

Outer element of case 1Outer element of case 2

0

1

2

3

4

Peak

hor

izon

tal t

ensil

e stre

ss (M

Pa)

2 3 4 51Monitoring section

Figure 19 Peak horizontal tensile stress of monitoring sections forroof in 2 cases

(2) After the adoption of wave converters the peakvertical tensile stress for inner element and the peakvertical compressive stress for outer element in thespan center drop remarkably which means that thepossibility of spalling damage for roof is reduced andthe impact load acting on the roof is decreased

(3) With wave converters the peak horizontal tensilestresses of inner elements for roof are generally lowerthan that of the traditional structure which couldreduce the amount of reinforcing bars

Conflicts of Interest

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

Acknowledgments

The authors gratefully acknowledge the support fromChongqing Graduate Student Innovation Project under

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

Grant no CYB14103 Chongqing Research Programof Basic Research and Frontier Technology underGrants nos cstc2014jcyjA30015 cstc2015 jcyjBX0073cstc2014jcyjA30014 and cstc2015 jcyjA30005 and Scienceand Technology Project of Land Resources and Real EstateManagement Bureau of Chongqing Government underGrant no CQGT-KJ-2014052

References

[1] V R Feldgun A V Kochetkov Y S Karinski and D ZYankelevsky ldquoBlast response of a lined cavity in a porous sat-urated soilrdquo International Journal of Impact Engineering vol 35no 9 pp 953ndash966 2008

[2] G-F Gao Y-C Li K Zhao and Y-C Pang ldquoDispersion andattenuation effects on stress waves in defense layer with cylin-drical shell embeddedrdquo Journal of Vibration and Shock vol 30no 12 pp 195ndash200 2011

[3] Z-L Wang J G Wang Y-C Li and C F Leung ldquoAttenuationeffect of artificial cavity on air-blast waves in an intelligentdefense layerrdquoComputers andGeotechnics vol 33 no 2 pp 132ndash141 2006

[4] Z W Liao Q J Liu and Z M Tian ldquoTests on the explosionresistance capacity of steel plate-polyurethane foam compositesandwich platesrdquo Chinese Journal of Underground Space andEngineering vol 1 no 3 pp 401ndash404 2005

[5] S Q Shi X J Zhang and P Yin ldquoStatic analysis of thenew defensive structure under explosive loadingrdquoUndergroundSpace vol 23 no 1 pp 66ndash68 2003

[6] G S Dhaliwal and G M Newaz ldquoEffect of layer structure ondynamic response and failure characteristics of carbon fiberreinforced aluminum laminates (CARALL)rdquo Journal of Dyn-amic Behavior of Materials vol 2 no 3 pp 399ndash409 2016

[7] H C He and D G Tang ldquoStudy on flexural resistance ofcomponent strengthened by carbon fiber reinforced plasticsunder explosive blastrdquo Journal of PLA University of Science andTechnology vol 3 no 6 pp 68ndash73 2002

[8] C J Montgomery R M Morison and D O Tutty ldquoDesignand construction of a buried precast prestressed concrete archrdquoPrecastPrestressed Concrete Institute Journal vol 38 no 1 pp40ndash57 1993

[9] SH Yang B Liang J C Gu J Shen andAMChen ldquoResearchon characteristics of prestress change of anchorage cable in anti-explosion model test of anchored cavernrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no s2 pp 3749ndash37562006

[10] S Kobielak T Krauthammer and A Walczak ldquoGround shockattenuation on a buried cylindrical structure by a barrierrdquo Shockand Vibration vol 14 no 5 pp 305ndash320 2007

[11] S Kobielak and T Krauthammer ldquoDynamic response of buriedsilo caused by underground explosionrdquo Shock and Vibrationvol 11 no 5-6 pp 665ndash684 2004

[12] V Yakushin L Belrsquokova and I Sevastyanova ldquoPropertiesof rigid polyurethane foams filled with glass microspheresrdquoMechanics of Composite Materials vol 48 no 5 pp 579ndash5862012

[13] A M Alhozaimy P Soroushian and F Mirza ldquoMechanicalproperties of polypropylene fiber reinforced concrete and theeffects of pozzolanic materialsrdquo Cement and Concrete Compos-ites vol 18 no 2 pp 85ndash92 1996

[14] A R Khaloo M Dehestani and P Rahmatabadi ldquoMechanicalproperties of concrete containing a high volume of tire-rubberparticlesrdquo Waste Management vol 28 no 12 pp 2472ndash24822008

[15] F Hernandez-Olivares and G Barluenga ldquoFire performanceof recycled rubber-filled high-strength concreterdquo Cement andConcrete Research vol 34 no 1 pp 109ndash117 2004

[16] S K Lim C S Tan O Y Lim and Y L Lee ldquoFresh andhardened properties of lightweight foamed concrete with palmoil fuel ash as fillerrdquo Construction and Building Materials vol46 no 3 pp 39ndash47 2013

[17] R Cortell ldquoApplication of the fourth-order Runge-Kuttamethod for the solution of high-order general initial value pro-blemsrdquoComputersamp Structures vol 49 no 5 pp 897ndash900 1993

[18] B S Desale and N R Dasre ldquoNumerical solution of the systemof six coupled nonlinear ODEs by Runge-Kutta fourth ordermethodrdquo Applied Mathematical Sciences vol 7 no 6 pp 287ndash305 2013

[19] J B Liu Y X Du and Q S Yan ldquoDynamic response ofunderground box structures subjected to blast loadrdquo Journal ofPLA University of Science and Technology vol 8 no 5 pp 520ndash524 2007

[20] B Zhang J Y Xu L Li and W Lin ldquoAnalysis of antidetona-tional property of foam concrete backfill layers in undergroundcompound structurerdquo Sichuan Building Science vol 36 no 6pp 135ndash138 2010

[21] X P Li J H Chen Y H Li and Y F Dai ldquoStudy of blastingseismic effects of underground chamber group in Xiluoduhydropower stationrdquo Chinese Journal of Rock Mechanics andEngineering vol 29 no 3 pp 493ndash501 2010

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of