Dynamic Model and Mechanical Properties of the Two ...

Research ArticleDynamic Model and Mechanical Properties ofthe Two Parallel-Connected CRLD Bolts Verified withthe Impact Tensile Test

Liangjun Hao,1,2 Weili Gong ,1,2 Manchao He,1,2 Yanqi Song,1,2 and JiongWang1,2

1State Key Laboratory for Geomechanics and Deep Underground Engineering, Beijing 100083, China2School of Mechanics and Civil Engineering, China University of Mining & Technology, Beijing 100083, China

Correspondence should be addressed to Weili Gong; [email protected]

Received 1 June 2018; Accepted 3 September 2018; Published 4 October 2018

Academic Editor: Mijia Yang

Copyright © 2018 Liangjun Hao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dynamic model was theoretically established for the two parallel-connected constant-resistance-large-deformation (CRLD) bolts,and the theoretical results were experimentally verifiedwith impact tensile tests on theCRLD bolts samples.The dynamic responsesof the double CRLD bolts were investigated under the impact loads with different intensities. The theoretical analyses showed that(1) under relatively small loading the CRLD bolts deform elastically and the deformation finally returns to zero and (2) under thehigh impact load, including the stable impact load and unstable impact load, the CRLD bolts export structural deformation afterthe initial elastic deformation. The deformation of the bolts eventually stabilizes at a certain amount of the elongation caused bythe relative sliding of the sleeves and rebars. The essential difference between the stable impact load and unstable impact load isthat, under the stable impact load, no structural deformation will occur after the impact load ends; under the unstable impact load,the structural deformation will still occur after the impact load ends. The obtained results are of theoretical implications for rocksupport design with CRLD bolts under the dynamical loading condition.

1. Introduction

With the increasing mining depths for both coal mines andmetal ore, tunnels being built in deep ground are oftensubjected to high stresses and complex geological conditions.Accordingly, surrounding rock masses in deep ground tendto experience various static and dynamic disasters such ascreep and rock bursts [1–4]. The constant-resistance-large-deformation bolt possesses more excellent performance thanthe traditional rebar in the areas of the control of roadwaywith slow and large deformation, the prevention of dynamicdisaster in deep mine, and the monitoring and early warningof landslide [5]. So far, there has been no a dynamic model forthe double CRLD bolts acting in parallel, which can be usedto explain the dynamic impact test results and recognize thedynamic response characteristics and furthermore providesguidance in the security and stability of the roadway support.

Over years, various ground support techniques and boltstructures have been developed for support and retention of

the newly exposed face surrounding the excavation in deepmines. Compared to the traditional strength bolt and yield-able bolt, the energy-absorbing bolt has both the strength ofthe former bolt and the deformation capacity of the latterone [6]. The energy-absorbing bolts that can meet the saidrequirement include the Garford Solid Dynamic Bolt [7, 8],Roffex [8], energy-absorbing rock bolt [9], cone bolt [10–12], MCB cone bolt (modified cone bolt with elongationas much as 180mm) [13, 14], and D bolt with large load-bearing and deformation (elongation as long as 400mm) [6].In order to adapt to the increasing demands for the supportsystem and bolt device in deep underground engineering, anovel bolt possessing ultrahigh energy-absorbing capacity bydeforming with an extraordinarily large elongation at highconstant resistance was developed by Manchao He, knownas the constant-resistance and large-deformation bolt (CRLDbolt in 2011) [15].

At present, the research progress on CRLD bolts mainlyfocuses on three aspects. (1) Theoretically, the constitutive

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 1423613, 17 pageshttps://doi.org/10.1155/2018/1423613

http://orcid.org/0000-0002-1441-5018

https://creativecommons.org/licenses/by/4.0/

https://doi.org/10.1155/2018/1423613

2 Mathematical Problems in Engineering

Anchored end Sleeve pipe Cone

Face pallet

Tightening nut

Shank rod

Constant-Resistance structureSeal grouting

Elongation

Figure 1: Schematic of the three-dimensional view for the CRLD bolt.

equation of the CRLD bolt under gradual loading is estab-lished, which proves the characteristics of constant resis-tance and large deformation of CRLD bolt [16]. (2) Interms of experimental studies, the CRLD bolt/cable statictensile tests verify the static constitutive equation [17] andin addition the CRLD bolt dynamic tensile tests embodythe energy-absorbing and anti-impacting performances andCRLD structural feature [18–21]. (3) In the project, the CRLDbolts are mainly used for the control of slow and largedeformation of deep soft rock roadway [5, 22, 23], roadwayeffective support under rock burst coal bump [2, 4–6, 9],and the monitoring and advance warning of the landslidein opencast mine [24, 25]. So far, there has been no atheoretically dynamic model of the double CRLDbolts actingin parallel for investigating the corresponding properties,except for experimental and engineering analysis [16].

In order to guide the support of the surrounding rockmass under the dynamic disasters in deep undergroundexcavation, the theoretical model for the double CRLD boltsacting in parallel under impact load is established accordingto the working principle and the impact tensile test of thedouble CRLD bolts support. After verifying the correctnessand stability of the theoretical model through experimentaldata, the dynamic response characteristics for the doubleCRLD bolts under the impact load with different intensitiesare further discussed.

2. Impact Tensile Test

2.1. Working Principle for Double CRLD Bolts Support.Figure 1 shows schematically the structure of the CRLDbolt which is actually a compound device consisting of thefollowing elements: a piston-like cone body installed on abolt shank (a rebar); a sleeve pipe with its inner diameterslightly smaller than the diameter of the large-end diameterof the cone; a face pallet and a tightening nut functioning asthe retention elements. The anchored end of the shank baris bonded by means of grout. When the axial external load(pull load) is applied on the far end (the face pallet end) of thebolt in the direction opposite to the anchored end, the sleevepipe will displace in the same direction relative to the conewhich is actually the elongation of theCRLDbolt.Themotionof the sleeve is equivalent to the displacement of the conerelative to the internal surface of the elastically deformablesleeve pipe. The small-end diameter of the cone is designedto be a little smaller than the internal diameter of the sleeve

in order to allow for the easy fitting of the cone body intothe pipe.The large-end diameter of the cone is slightly largerthan the inner diameter of the sleeve pipe in order to generatethe frictional resistance (i.e., the working resistance of theCRLD bolt) during the relative sliding between the cone andsleeve pipe.The elastic limit of the shank bar is more than thefrictional resistance. Thus the shank only deforms elasticallywhen the bolt is subjected to the external load.

Figure 2 demonstrates the working principle of the dou-ble CRLD bolts acting in parallel in stabilizing the surround-ing rockmass. It is considered that the double CRLDbolts areequivalent with the same geometric parameters and materialparameters and maintain simultaneous deformation duringthe support process. The anchored end is fixed inside thestable interior region by resin or cement grouts and the sleeveis located in the unstable surface region. The face pallet is inclose contact with the free surface of the surrounding rockmass and plays a supporting role on the roadway. During thedilation of the surrounding rock mass, the bolts system willrestrain the dilation so that a tensile load is induced at the facepallet end.

In the initial support process, the pull force exerted bythe pallet and nut on the sleeve pipe does not reach theconstant resistance force 𝑃𝑐𝑟 which is the maximum staticfriction; when the cone slides inside the sleeve pipe, the sleevepipe, being relatively static with the cone in the directionopposite to anchored end of the bar, only exports elasticelongations of the shank rod as shown in Figure 2(a). If thepull force exerted by the pallet and nut on the sleeve pipeexceeds the constant resistance force 𝑃𝑐𝑟, the sleeve pipe willbe displaced in the direction opposite to anchored end ofthe bar, exporting elongations with structural deformation asshown in Figure 2(b).

2.2. Testing System andResults. Figure 3 illustrates the impacttensile test system of the double CRLD bolts acting inparallel (i.e., modified SHTB test system developed at StateKey Laboratory for Geomechanics and Deep UndergroundEngineering in China University of Mining & TechnologyBeijing), which is specially used to explore the dynamiccharacteristics of the double bolts acting in parallel underimpact load. The double CRLD bolts impact tensile test sys-tem is mainly divided into three parts: Hopkinson dynamicloading system (Figure 3(a)(i)), double bolts impact ten-sile system (Figure 3(a)(ii)), and data acquisition system(Figure 3(a)(iii)).

Mathematical Problems in Engineering 3

Stableinterior region

Unstablesurface region

Structural deformationwith cone sliding inside sleeve

Grout

Fixed end(a)

(b)

m

m

M

T

T

P(t)

Shank bar Cone

Sleeve pipe Face pallet

m

m

MP(t)

T

T

Elastic deformation with shank elongating

Figure 2: Working principle of the double CRLD bolts acting in parallel in stabilizing the surrounding rock mass: (a) initial support withelastic deformation of the double bolts and (b) late support with structural deformation of the double bolts.

Table 1: Material parameters of the CRLD bolt samples for experiment.

Rod and cone Sleeve PalletMaterial #45 steel #20 Ti steel SteelMass (Kg) 3.26 1.24 33.2Elastic modulus (GPa) 210 206 210Poisson’s ratio 0.269 0.3 0.3

Specifically the Hopkinson dynamic loading systemcontains the following sections: a loading system beingable to provide pressure range between 0 and 25MPa(Figure 3(b)(i)), a guidance system with length 3950mm(Figure 3(b)(ii)), and a bullet made of #35 CrMn steel (length𝐿 is 800mm, diameter is 75mm, and quality is 27.6Kg)as shown in Figure 3(b)(iii). In detail double bolts impacttensile system consists of the following elements: an impactrod whose diameter is equal to that of the bullet (lengthis 1500mm, quality is 51.7Kg, density is 7800Kg/m3, andelastic stress wave velocity is 5190 m/s), two guide frameswiththe goal of supporting the impact rod coaxial with theloading system, and a fixed bearing functioning to fix theanchored ends of the double CRLD bolts (the guide framesand fixed bearing are all made of high-strength cast steel),two equivalent CRLD bolts samples with the same geometricparameters and material parameters (the parameters arepresented in Figure 4 and Table 1), the face pallet being ableto slide freely in the horizontal direction and involving threeforce sensors to measure impact load when impact rod ishit, and the tightening nut fixing the face pallet over thesleeve. Figure 3(b)(iv, viii) shows the double CRLD boltsimpact tensile system from different perspectives. The data

acquisition system includes speed monitor measuring thevelocity of the bullet rushing out of the bore (Figure 3(b)(v)),load monitor real-time displaying impact load exerted bythe impact rod on the face pallet (Figure 3(b)(vi)), anddisplacement monitor capturing the elongation of sleeve tailend in time (Figure 3(b)(vii)).

Figure 3(c) demonstrates the schematic view of theimpact tensile test of the double CRLD bolts acting in parallelin the vertical direction which is more convenient to under-stand the experimental process. After the erection of all theexperimental apparatus is completed, loading controller setsthe barometric pressure, and simultaneously the speed, load,and displacement monitor are ready to start sampling. Thepressure released by air cylinder pushes forward the bullet todo accelerated movement in the bore. The high-speed bulletthat rushes out of the bore strikes the impact rod whoseopposite end intimately contacts the face pallet, so the doubleCRLDboltswhose sleeve pipes are bonded to the face pallet inparallel by their nuts produce the corresponding mechanicalbehavior. During this process speed monitor records thevelocity at which the bullet hits the impact rod, load monitorreal-time records and images the load on the face palletthrough the force sensors, and simultaneously displacement


Schematic view in the horizontal direction

i. Dynamic loading system ii. Double bolts impact tensile system

Fixed rodCRLD

bolt

Guide frame

Measuring velocity

Measuring load

Measuring elongation

iii. Data acquisition system

BoreAir cylinder

Bullet

Fixed bearing

Impact rod

Joint pipe

Force sensors

Protecting mask

(a)

i. Air cylinder ii. Bore iii. Bullet iv. Impact system

v. Speed monitor vi. Load monitor vii. Displacement monitor

viii. Impact system

Experimental instruments

(b)

Schematic view in the vertical direction

Face pallet

First CRLD bolt

Second CRLD bolt

Fixed bearing

Impact rodAir cylinder

Bullet

Fixed rod

(c)

Figure 3: Experimental setup for the impact tensile test of the double CRLD bolts: (a) schematic view in the horizontal direction; (b)experimental instruments; and (c) schematic view in the vertical direction.

Φ 22mm Φ 33mmΦ 25mm

Φ 24mm

Φ 165mm 270 mm

430mm

394mm

500mm 70mm

Φ 22mm

Figure 4: Geometric parameters of the double CRLD bolts samples for experiment.


(0.30, 271.06)

Stable value0 KN

(1.97, 140.81)

1 2 3 4 5 6 7 8 9 10 110Time (ms)

−50

0

50

100

150

200

250

300

Load

(KN

)

Figure 5: Impact load-time curve for double CRLD bolts samplesacting in parallel. (The gas pressure is 1.0MPa and the speed of thebullet is 6.14m/s.)

monitor remembers and displays the elongation of the doublebolts tails.

Figure 5 shows the experimental impact load-time curvefor the double CRLD bolts samples in parallel. Intensity ofair source is applied at 1.0MPa and the strike speed of thebullet is measured as 6.14m/s. It can be seen that the loaddrastically attains the maximum value 271.06KNat amoment0.30ms in the time axis and sharply drops to the stable valueimmediately. Interestingly, the load reaches a second peakvalue 140.81KN when at 1.97ms.

3. Dynamic Model for Double CRLD BoltsActing in Parallel

The impact load is simulated by a trigonometric function forits mechanical properties of rapid rise and fall in a very shortperiod of time, and it can be expressed by

𝑃 (𝑡) = 𝑃𝑚 [1 + sin (𝜆𝑡 − 𝜋2 )] (1)

where 𝑃𝑚 and 𝜆 determine the intensity and duration ofimpact load, respectively. Specifically, the intensity of impactload (denoted by 𝑃𝑚𝑎𝑥) is 𝑃𝑚𝑎𝑥 = 2𝑃𝑚, and the duration ofimpact load (denoted by Δ𝑡) is Δ𝑡 = 2𝜋/𝜆. It can be seen thatthe impact load achieves the maximum value at the time of𝜋/𝜆 from the curve of impact load as shown in Figure 6.

When the intensity of impact load varies, the forcecondition and motion situation of the CRLD bolts havegreat difference. Hence, the impact load is divided into threetypes in accordance with the different intensities. (i) For theelastic impact load, the CRLD bolts only deform elasticallyduring the entire process. (ii) For the stable impact load, thestructural deformation of the CRLD bolts is generated; thatis, the sleeve slides relatively to the cone. Notably, the relativeslippage has stopped when the impact load ends. (iii) Forunstable impact load, the CRLD bolts also produce structuraldeformation, but the relative sliding will continue after theimpact load is over. The force and deformation of the CRLD

P

Pmax

oΔt

2

Δt t

P = Pm[1 + ＭＣＨ(t−

2)]

Pmax = 2Pm

Δt =2

Figure 6: The curve of impact load over time (𝑃𝑚𝑎𝑥 and Δ𝑡 are theintensity and duration of impact load, respectively).

bolts under the three kinds of impact load are discussed, andthe mechanical properties are further studied.

Figure 7 clearly explains the mechanical model of thedouble CRLD bolts acting in parallel under the elastic impactload. The sleeve and the shank rod always have no relativesliding, and only the shank rod is elastically elongated andcompressed. The motion of the double CRLD bolts consistsof two processes: the combined action of the impact loadand the internal force of the shank (see Figure 7(a)); then theinternal force of the shank alone (see Figure 7(b)). The boltsare in a state of rest at the initial moment, and the impactload begins to act on the face pallet, as shown in Figure 7(a).Under the impact load and the internal force of the shank, thedouble bolts elastically deform, as shown in Figure 7(b). Afterthe impact load is complete, the bolts oscillate elastically onlyunder the shank force, as shown in Figure 7(c).

The impact load 𝑃(𝑡) is exerted on the face pallet alongthe direction of movement (i.e., t ≤ Δt), and the internalforces 2𝑘𝑥 (𝑘 is the stiffness of the single shank) of the doubleshank rods are applied in the opposite direction of motion, asshown in Figure 7(b). The displacement of the double bolts,starting with the initial rest position, is represented by 𝑥. Aforce balance on the double bolts gives

(𝑀 + 2𝑚) �� = 𝑃 (𝑡) − 2𝑘𝑥 (2)

where 𝑚 is the mass of the shank and cone for the singleCRLD bolt and 𝑀 is the mass of the face pallet and doublesleeves. The overdot denotes differentiation with respect tothe time, 𝑡.

Substituting (1) into (2) then can be rewritten into astandard form of second-order ordinary nonhomogeneousdifferential equation.

(𝑀 + 2𝑚) �� + 2𝑘𝑥 = 𝑃𝑚 sin (𝜆𝑡 − 𝜋2 ) + 𝑃𝑚 (3)


m

m

M

Original state

k

k

T=0

T=0

P(t)(a)

Shank bar Cone

Sleeve pipeFace pallet

m

m

M

k

k

T=kx

T=kx

P(t)(b)

ox

m

m

M

k

k

T=kx

T=kx

P(t)=0(c)

ox

Nomenclaturem - mass of single bolt shank and coneM - mass of double sleeve pipe and face palletT - elastic force of the shank barP - impact load

t - duration of the impact loadk - stiffness of the shank barx - elongation with the elastic deformation

Combined action of the impact load and the elastic force of the shank rod

Impact load turning into zero and only the elastic force of the shank rod

Pmax - intensity of the impact load

ending at =

≤

>

= =

P

Pmax

o Δt

2

Δt t

Figure 7: Mechanical model of double CRLD bolts under the elastic impact load: (a) original state of the double bolts; (b) combined actionof the impact load and the elastic force of the shank bar; and (c) only the elastic force of the shank bar.

The solution to (3), with the introduction of two freevariables 𝐶1 and 𝐶2, is

𝑥 = 𝐶1 sin (��𝑡) + 𝐶2 cos (��𝑡)+ 𝑃𝑚2𝑘 − (𝑀 + 2𝑚) 𝜆2 sin (𝜆𝑡 − 𝜋2 ) + 𝑃𝑚2𝑘

(4)

�� = 𝐶1�� cos (��𝑡) − 𝐶2�� sin (��𝑡)+ 𝑃𝑚𝜆2𝑘 − (𝑀 + 2𝑚) 𝜆2 cos (𝜆𝑡 − 𝜋2 ) (5)

where

�� = √ 2𝑘𝑀 + 2𝑚. (6)

For the initial conditions 𝑥 = �� = 0 when 𝑡 = 0, the twofree variables can be given by the following.

𝐶1 = 0𝐶2 = 𝑃𝑚2𝑘 − (𝑀 + 2𝑚)𝜆2 − 𝑃𝑚2𝑘

(7)

At the time of impact load ending, that is, t = Δt, thedisplacement and velocity of the bolts, respectively, become𝑥Δ𝑡 = 𝑥(Δ𝑡) and ��Δ𝑡 = ��(Δ𝑡).

After the impact load is completed (i.e., t > Δt), the boltscontinue to elastically deform under the internal forces of the


shanks as shown in Figure 7(c). Another force balance givesthe following.

(𝑀 + 2𝑚) �� = −2𝑘𝑥 (8)

The solution to (8), with the two free variables 𝐶3 and 𝐶4,is as follows.

𝑥 = 𝐶3 sin [�� (𝑡 − Δ𝑡)] + 𝐶4 cos [�� (𝑡 − Δ𝑡)] (9)

�� = 𝐶3�� cos [�� (𝑡 − Δ𝑡)] − 𝐶4�� sin [�� (𝑡 − Δ𝑡)] (10)

Substituting the continuity conditions 𝑥 = 𝑥Δ𝑡 and �� =��Δ𝑡 when 𝑡 = Δ𝑡 into (9) and (10), then the two free variablescan be given by the following.

𝐶3 = ��Δ𝑡��𝐶4 = 𝑥Δ𝑡

(11)

Figure 8 clearly explains the mechanical model of thedouble CRLD bolts acting in parallel under the stable impactload. The double CRLD bolts not only generate elasticdeformation but also cause structural deformation.The sleevepipes slide with respect to the cones, and the relative slidinghas stopped when the impact load ends. The situationof force and motion for the double CRLD bolts can bedivided into four processes under the stable impact load: theelastic deformation (Figure 8(b)); the structural deformation(Figure 8(c)); the elastic recovery under the impact load andshank force (Figure 8(d)); and the elastic recovery under onlythe shank force (Figure 8(e)).

At the initial moment, the double bolts are at a state of restand the impact load begins to act on the face pallet, as shownin Figure 8(a). The double bolts deform elastically underthe impact load and shank force, as shown in Figure 8(b).Once the amount of the deformation exceeds the maximumelastic deformation of the CRLD bolt, implying that theinternal force of the shank exceeds the constant resistance ofthe CRLD bolt, the double bolts start to undergo structuraldeformation. The sleeves and the cones slide relative to eachother.The sleeves continue tomove forward under the impactload and sliding friction, while the cones oscillate back andforth under the shank force and sliding friction, as shown inFigure 8(c).The speed of the sleeves gradually decreases sincethe impact load on the sleeves gradually decreases at a laterstage and the sliding friction remains unchanged. When thespeed of the sleeve pipes is reduced to zero, the relative slidingis completed. At this moment, the impact load continues towork.The double CRLD bolts produce elastic elongation andcompression under the impact load and shank force until theend of the impact load, as shown in Figure 8(d). After theimpact load is over, the bolts continue to deform elasticallyunder the internal force of the shank along, as shown inFigure 8(e).

In the initially elastic deformation stage (i.e., 𝑥 ≤ 𝑥𝑐𝑟,and 𝑥𝑐𝑟 is the amount of maximum elastic deformation ofthe CRLD bolt corresponding to the constant resistance),as shown in Figure 8(b), the double bolts elastically deformunder the impact load and the shank force. Therefore,

the force balance equation is equivalent to (2), and thedisplacement-time relationship and velocity-time relation-ship of the bolts in initially elastic stage under stable impactload are (4) and (5), respectively.

Under the elastic impact load, the bolts do not reachthe maximum elastic deformation during the whole impactprocess. Different from the elastic impact load, the bolts reachthe amount 𝑥𝑐𝑟 of maximum elastic deformation at the end ofthe initial phase under the stable impact load. It can be seenthat the end time of the initial phase (denoted by 𝑡1) satisfies𝑥(𝑡1) = 𝑥1 = 𝑥𝑐𝑟. The maximum elastic deformation is givenby (for details see [16])

𝑥𝑐𝑟 = 𝑃𝑐𝑟𝑘 (12)

𝑃𝑐𝑟 = 2𝜋𝐼𝑠𝐼𝑐𝑓 (13)

where 𝑃𝑐𝑟 is the constant resistance of the CRLD bolt, 𝐼𝑠 isthe sleeve elastic constant, 𝐼𝑐 is the cone geometrical constant,and 𝑓 is the coefficient of the static friction.

After the initial phase is completed, the speed of sleeveand shank becomes as follows.

��1 = 𝑥 (𝑡1)= [𝑃𝑚2𝑘 − 𝑃𝑚2𝑘 − (𝑀 + 2𝑚) 𝜆2 ] �� sin (��𝑡1)

+ 𝑃𝑚𝜆2𝑘 − (𝑀 + 2𝑚) 𝜆2 cos(𝜆𝑡1 − 𝜋2 )(14)

Once the bolts deform beyond the maximum elasticdeformation, implying that the internal force in the shank rodexceeds the constant resistance, the bolts begin to producestructural deformation by the relative sliding of the sleeveand shank, as shown in Figure 8(c). The force condition andmovement situation become different as the sleeve and shankare no longer relatively stationary. For the sleeves and facepallet, the impact load is exerted in the direction ofmovementand the sliding friction is applied in the opposite direction,satisfying the equation

𝑀�� = 𝑃 (𝑡) − 2𝐹𝑑 (15)

where 𝐹𝑑 is the sliding friction force between the shankand sleeve and given by (for details see [16])

𝐹𝑑 = 2𝜋𝐼𝑠𝐼𝑐𝑓𝑑 (16)

where 𝑓𝑑 is the coefficient of dynamic friction betweenthem. Solving (15), we can get the displacement-time relation-ship and speed-time relationship of the bolts in the structuraldeformation stage under stable impact load:

𝑥 = − 𝑃𝑚𝑀𝜆2 sin (𝜆𝑡 − 𝜋2 ) + 𝑃𝑚 − 2𝐹𝑑2𝑀 𝑡2 + 𝐶5𝑡 + 𝐶6 (17)

�� = − 𝑃𝑚𝑀𝜆 cos(𝜆𝑡 − 𝜋2 ) + 𝑃𝑚 − 2𝐹𝑑𝑀 𝑡 + 𝐶5 (18)

where 𝐶5 and 𝐶6 are two free variables. Substituting thecontinuity conditions 𝑥 = 𝑥1 and 𝑥 = 𝑥1 when 𝑡 = 𝑡1 into


m

m

MOriginal state

k

k

T=0

T=0

(a)

Shank bar Cone


m

m

Mk

k

T=kx

T=kx

(b)

o

m

m

Mk

k(c)

o

Elongation with only elastic deformation of the shank bar

Elongation with structural deformation (i.e. relative sliding of the cone and

o

m

m

Mk

kP(t)

P(t)

P(t)

(d)

o x

x

x

xm

o

Joint action of the impact load and the elastic force of the shank rod

m

m

Mk

kP(t)=0

P(t)=0

(e)

o x

o

Impact load turning into zero and only the elastic force of the shank rod


k - stiffness of the shank barx - elongation with the elastic deformation

- duration of the impact load

– maximum elastic deformation of the bolt

Pmax- intensity of the impact load = =

>

≤

ending at =

≤

>

=sleeve pipe) ending at = and

P

Pmax

o tΔt

2

Δt

=(--+)

=(--+)

=(--+)

=(--+)

=(-)

=(-)

+

+

Figure 8: Mechanical model of double CRLD bolts under the stable impact load: (a) original state of the double bolts; (b) initially elasticdeformation stage; (c) structural deformation stage; (d) late elastic oscillation stage under the combined action of the impact load and theelastic force of the shank bar; and (e) late elastic oscillation stage only under the elastic force of the shank bar.


(17) and (18), then the two free variables can be obtained asfollows.

𝐶5 = ��1 + 𝑃𝑚𝑀𝜆 cos (𝜆𝑡1 − 𝜋2 ) − 𝑃𝑚 − 2𝐹𝑑𝑀 𝑡1𝐶6 = 𝑥1 + 𝑃𝑚𝑀𝜆2 sin (𝜆𝑡1 − 𝜋2 ) − 𝑃𝑚 − 2𝐹𝑑2𝑀 𝑡2

1− 𝐶5𝑡1

(19)

When the velocity of the sleeves is reduced to zero, thesleeves and the shanks stop sliding relative to each other,and the bolts no longer produce structural deformation. Itcan be seen that the end time of the structural deformationphase, denoted by 𝑡2, satisfies ��(𝑡2) = 0. At this time, thedisplacement of the sleeves and face pallet, denoted by 𝑥2,becomes 𝑥2 = 𝑥(𝑡2).

On the other hand, the shanks and cones oscillate backand forth under the shank forces and sliding friction duringthe structural deformation phase. The force balance on thecones gives

𝑚��𝑚 = 𝑘 (𝑥1 − 𝑥𝑚) − 𝐹𝑑 (20)

where 𝑥𝑚 is the displacement of the cones during thestructural deformation stage starting from the end of theinitial elastic phase. Substituting (12), (13), and (16) into (20),then we can have the following.

𝑚��𝑚 + 𝑘𝑥𝑚 = 2𝜋𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑) (21)

It is convenient to use the elapsed time 𝜏 since theinitial movement of the cones and shanks as the independentvariable; that is, 𝜏 = 𝑡 − 𝑡1, so the solution to (21) is

𝑥𝑚 = 𝐶7 sin (𝜔𝜏) + 𝐶8 cos (𝜔𝜏) + 2𝜋𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑)𝑘 (22)

��𝑚 = 𝐶7𝜔 cos (𝜔𝜏) − 𝐶8𝜔 sin (𝜔𝜏) (23)

where 𝐶7 and 𝐶8 are two free variables and 𝜔 = √𝑘/𝑚.Substituting the initial conditions 𝑥𝑚 = 0 and ��𝑚 = 𝑥1 when𝜏 = 0 into (22) and (23), then the two free variables can begiven by the following.

𝐶7 = 𝑥1𝜔𝐶8 = −2𝜋𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑)𝑘

(24)

Substitute 𝜏 = 𝑡−𝑡1 into (22) and (23); then the expressionof the displacement of the cone with respect to the variable 𝑡will be

𝑥𝑚 = ��1𝜔 sin [𝜔 (𝑡 − 𝑡1)]− 2𝜋𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑)𝑘 cos [𝜔 (𝑡 − 𝑡1)]+ 2𝜋𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑)𝑘

(25)

and the relationship between the velocity of the cone andthe time 𝑡 will be as follows.

��𝑚 = ��1 cos [𝜔 (𝑡 − 𝑡1)]+ 2𝜋𝜔𝐼𝑠𝐼𝑐 (𝑓 − 𝑓𝑑)𝑘 sin [𝜔 (𝑡 − 𝑡1)] (26)

At the end time of the structural deformation, thedisplacement of the cone (denoted by 𝑥𝑚) becomes

𝑥𝑚 = 𝑥𝑚 (𝑡2) (27)

and the speed of the cone (denoted by V𝑚) changes intothe following.

V𝑚 = ��𝑚 (𝑡2) (28)

When the speed of the sleeves is reduced to zero, thesleeves and shanks stop sliding relative to each other andbegin tomove together. According to the law of conservationof momentum, the common velocity (denoted by V𝑀+𝑚) ofthem satisfies the following.

2𝑚V𝑚 = (𝑀 + 2𝑚) V𝑀+𝑚 (29)

The solution to (29) is as follows.

V𝑀+𝑚 = 2𝑚𝑀 + 2𝑚V𝑚 (30)

Under the stable impact load, the impact load still existsat the end of the structural deformation of the bolts; that is,𝑡2 ≤ Δ𝑡. The double CRLD bolts oscillate elastically underthe impact load and shank force, as shown in Figure 8(d). Let𝑥𝑀+𝑚 denote the common displacement of the sleeves andshanks during the elastic oscillation starting from the end ofthe structural deformation. The force balance gives

(𝑀 + 2𝑚) ��𝑀+𝑚 = 2𝑘 (𝑥1 − 𝑥𝑚 − 𝑥𝑀+𝑚) − 𝑃 (𝑡) (31)

Substituting (1) into (31), it can be rewritten into astandard form of second-order ordinary nonhomogeneousdifferential equation.

(𝑀 + 2𝑚) ��𝑀+𝑚 + 2𝑘𝑥𝑀+𝑚= 2𝑘 (𝑥1 − 𝑥𝑚) − 𝑃𝑚 [1 + sin (𝜆𝑡 − 𝜋2 )] (32)

Solving (32), we can obtain the displacement-time rela-tionship and speed-time relationship of the bolts during theelastic oscillation under the impact load and shank force

𝑥𝑀+𝑚 = 𝐶9 sin [�� (𝑡 − 𝑡2)] + 𝐶10 cos [�� (𝑡 − 𝑡2)]+ 𝑃𝑚(𝑀 + 2𝑚) 𝜆2 − 2𝑘 sin (𝜆𝑡 − 𝜋2 )+ 2𝑘 (𝑥1 − 𝑥𝑚) − 𝑃𝑚2𝑘

(33)

��𝑀+𝑚 = 𝐶9�� cos [�� (𝑡 − 𝑡2)] − 𝐶10�� sin [�� (𝑡 − 𝑡2)]+ 𝑃𝑚𝜆(𝑀 + 2𝑚) 𝜆2 − 2𝑘 cos (𝜆𝑡 − 𝜋2 ) (34)


where 𝐶9 and 𝐶10 are two free variables. Substituting thecontinuity conditions 𝑥𝑀+𝑚 = 0 and ��𝑀+𝑚 = V𝑀+𝑚 when𝑡 = 𝑡2 into (33) and (34), then the two free variables can beobtained by the following.

𝐶9 = 1𝜔 [V𝑀+𝑚 − 𝑃𝑚𝜆(𝑀 + 2𝑚)𝜆2 − 2𝑘 cos (𝜆𝑡2 − 𝜋2 )]𝐶10

= − 𝑃𝑚(𝑀 + 2𝑚) 𝜆2 − 2𝑘 sin (𝜆𝑡2 − 𝜋2 )− 2𝑘 (𝑥1 − 𝑥𝑚) − 𝑃𝑚2𝑘

(35)

When the impact load is over (𝑡 = Δ𝑡), the commondisplacement of the sleeves and the shanks becomes

𝑥𝑀+𝑚Δ𝑡

= 𝑥𝑀+𝑚 (Δ𝑡) (36)

and their common velocity turns into the following.

𝑥𝑀+𝑚Δ𝑡

= ��𝑀+𝑚 (Δ𝑡) (37)

At the subsequent times (i.e., 𝑡 > Δ𝑡), the sleeves andshanks continue to oscillate elastically only under the internalforces of the shanks, satisfying the following equation.

(𝑀 + 2𝑚) ��𝑀+𝑚 = 2𝑘 (𝑥1 − 𝑥𝑚 − 𝑥𝑀+𝑚) (38)

The solution to (38), with two free variables 𝐶11 and 𝐶12,is as follows.

𝑥𝑀+𝑚 = 𝐶11 sin [�� (𝑡 − Δ𝑡)] + 𝐶12 cos [�� (𝑡 − Δ𝑡)]+ (𝑥1 − 𝑥𝑚) (39)

��𝑀+𝑚 = 𝐶11�� cos [�� (𝑡 − Δ𝑡)] − 𝐶12�� sin [�� (𝑡 − Δ𝑡)] (40)

Substituting the continuity conditions 𝑥𝑀+𝑚 = 𝑥𝑀+𝑚Δ𝑡

and��𝑀+𝑚 = ��𝑀+𝑚Δ𝑡

when 𝑡 = Δ𝑡 into (39) and (40), then the twofree variables can be obtained as follows.

𝐶11 = ��𝑀+𝑚Δ𝑡��

𝐶12 = 𝑥𝑀+𝑚Δ𝑡

− (𝑥1 − 𝑥𝑚)(41)

As a further result, the displacement 𝑥 of the bolts duringthe elastic oscillation relative to the initial position of thewhole process can be given by the following.

𝑥 = 𝑥2 − 𝑥𝑀+𝑚 (42)

Figure 9 clearly explains the mechanical model of thedouble CRLD bolts acting in parallel under the unstableimpact load. The double bolts also produce structural defor-mation, and the sleeves and shanks slide relative to each other.However, unlike under the stable impact load, the structuraldeformation caused by relative sliding continues to occurafter the impact load is completed. The situation of force andmotion for the double CRLD bolts can also be divided intofour processes under the unstable impact load: the elastic

deformation (Figure 9(b)); the structural deformation underthe impact load and shank forces (Figure 9(c)); the structuraldeformation under the shank forces alone (Figure 9(d)); andthe elastic recovery (Figure 9(e)).

At the initial moment, the double bolts are at a state of restand the impact load begins to act on the face pallet, as shownin Figure 9(a). The double bolts deform elastically underthe impact load and shank forces, as shown in Figure 9(b).Once the amount of the deformation exceeds the maximumelastic deformation of the CRLD bolt, implying that theinternal force in the shank exceeds the constant resistance ofthe CRLD bolt, the double bolts start to undergo structuraldeformation. The sleeves and the cones slid relative to eachother.The sleeves continue tomove forward under the impactload and the sliding friction, while the cones oscillate backand forth under the shank forces and the sliding friction,as shown in Figure 9(c). When the impact load is over, therelative movement of the sleeves and cones does not stop,and the structural deformation continues to occur underonly the shank forces. The sleeves still move forward underthe sliding friction, while the cones still oscillate back andforth, as shown in Figure 9(d). The speed of the sleeve pipesgradually decreases due to the sliding friction. When thespeed is reduced to zero, the structural deformation caused bythe relative sliding no longer occurs. The sleeves and shanksbegin to recover elastically under the internal forces of theshanks, as shown in Figure 9(e).

In the initially elastic deformation stage (i.e., 𝑥 ≤ 𝑥𝑐𝑟),as shown in Figure 9(b), the double bolts elastically deformunder the unstable impact load and the shank forces. It iscompletely equivalent to the initial stage under the stableimpact load.

Once the bolts deform beyond the maximum elasticdeformation 𝑥𝑐𝑟, meaning that the internal force in the shankrod exceeds the constant resistance 𝑃𝑐𝑟, the bolts begin toproduce structural deformation by the relative sliding of thesleeve and shank (Figure 9(c)). The situation of the force andmotion becomes different as the sleeve and shank are nolonger relatively stationary. For the sleeves and face pallet,the impact load is exerted in the direction of movementand the sliding friction is applied in the opposite direction.Therefore, the force balance equation is equivalent to (15),and the displacement-time relationship and velocity-timerelationship of the bolts in the structural deformation stageunder the unstable impact load and shank forces are (17) and(18), respectively.

Under the stable impact load, the structural deformationis completed before the end of the impact load. Differently,under the unstable impact load, the structural deformationcontinues to occur after the impact load is over. At the timeof the impact load ending, i.e., 𝑡 = Δ𝑡, the displacement of thesleeves and pallet changes into𝑥Δ𝑡 = 𝑥 (Δ𝑡)

= 𝑃𝑚𝑀𝜆2 + 𝑃𝑚 − 2𝐹𝑑2𝑀 (2𝜋𝜆 )2 + 𝐶5 2𝜋𝜆 + 𝐶6 (43)

and the velocity of them turns into the following.

��Δ𝑡 = �� (Δ𝑡) = 𝑃𝑚 − 2𝐹𝑑𝑀 2𝜋𝜆 + 𝐶5 (44)


m

m

M

Original state

k

k

T=0

T=0

(a)

Shank bar Cone


m

m

Mk

k

T=kx

T=kx

(b)

o

m

m

Mk

k(c)

o

Elongation with only elastic deformation of the shank bar

Elongation with structural deformation under only the elastic force of the shank

o

(d)

o

k

k(e)

o

o

Elastic oscillation of the fact pallet and the cone together

o

m

m

Mk

k

Elongation with structural deformation under the impact load and the elastic

m

m

M


k - stiffness of the shank barx - elongation with the elastic deformation

- duration of the impact load

– maximum elastic deformation of the bolt

Pmax- intensity of the impact load()=

()=

()=

()

()

=(-)

=(-)

=(-)

=(-)

+

x

xm

xm

x

x

bar ending at = and =

>

force of the shank bar ending at =

> and ≤

ending at =

≤

= =

P

Pmax

otΔt

2

Δt

=(--+)

=(--+)

Figure 9: Mechanical model of double CRLD bolts under the unstable impact load: (a) original state of the double bolts; (b) initially elasticdeformation stage; (c) structural deformation stage under the combined action of the impact load and the elastic force of the shank rod; (d)structural deformation stage only under the elastic force of the shank bar; and (e) late elastic oscillation stage.


After the impact load is over, i.e., 𝑡 > Δ𝑡, the sleeves con-tinue to slide forward with respect to the shanks resulting inthe structural deformation (Figure 9(d)). At the subsequenttimes, the sleeves and pallet under only the sliding frictionsatisfy the following equation.

𝑀�� = −2𝐹𝑑 (45)

Solving (45), we can obtain the displacement-time rela-tionship and speed-time relationship of the sleeves after theimpact load ending:

𝑥 = −𝐹𝑑𝑀𝑡2 + 𝐶13𝑡 + 𝐶14 (46)

�� = −2𝐹𝑑𝑀 𝑡 + 𝐶13 (47)

where 𝐶13 and 𝐶14 are two free variables. Substituting thecontinuity conditions 𝑥 = 𝑥Δ𝑡 and �� = ��Δ𝑡 when 𝑡 = Δ𝑡into (46) and (47), then the two variables can be given by thefollowing.

𝐶13 = ��Δ𝑡 + 2𝐹𝑑𝑀 Δ𝑡𝐶14 = 𝑥Δ𝑡 + 𝐹𝑑𝑀Δ𝑡2 − 𝐶13Δ𝑡

(48)

The velocity of the sleeves gradually decreases due to thesliding friction in the opposite direction of the movement.When the speed is reduced to zero, the sleeves and shanksstop sliding relative to each other, and the bolts no longerproduce structural deformation. It can be seen that the endtime of the structural deformation, denoted by 𝑡2, satisfies��(𝑡2) = 0. At this time, the displacement of the sleeves,denoted by 𝑥2, becomes 𝑥2 = 𝑥(𝑡2).

During the entire process of the structural deformation,including the deformation under the impact load and shankforces (Figure 9(c)) as well as the deformation under onlythe shank forces (Figure 9(d)), the shanks and cones alwaysoscillate back and forth under the internal forces of theshanks and sliding frictions. It is equivalent to the structuraldeformation phase under the stable impact load. Similarly,𝑥𝑚is used to represent the displacement of the cones during thewhole process of structural deformation starting from the endof the initial elastic phase, and 𝜏 = 𝑡 − 𝑡1. It can be seen thatthe displacement of the cones satisfies (20).The displacement𝑥𝑚 and the velocity ��𝑚 of the cones with respect to the timevariable 𝑡 are, respectively, (25) and (26).

At the end time of the structural deformation caused bythe relative sliding of the sleeves and cones, the displacementand velocity of the cones are, respectively, (27) and (28). Itneeds special attention that 𝑡2 > Δ𝑡 under unstable impactload and 𝑡2 ≤ Δ𝑡 under the stable impact load. This isthe essential difference between the unstable impact loadand stable impact load: under the unstable impact load, thestructural deformation will still occur after the impact loadends; under the stable impact load, no structural deformationwill occur after the impact load ends.

After the structural deformation of the bolts is completed,the bolts oscillate elastically under only the internal forces

of the shank rods (Figure 9(e)). Similarly, let 𝑥𝑀+𝑚 denotethe common displacement of the sleeves and shanks duringthe elastic oscillation starting from the end of the structuraldeformation. The force balance of the bolts gives the follow-ing.

(𝑀 + 2𝑚) 𝑥𝑀+𝑚 = 2𝑘 (𝑥1 − 𝑥𝑚 − 𝑥𝑀+𝑚) (49)

Solving (49), we can obtain the displacement-time rela-tionship and speed-time relationship of the bolts during theelastic oscillation under the unstable impact load and shanksforces

𝑥𝑀+𝑚 = 𝐶15 sin [�� (𝑡 − 𝑡2)] + 𝐶16 cos [�� (𝑡 − 𝑡2)]+ (𝑥1 − 𝑥𝑚) (50)

��𝑀+𝑚 = 𝐶15�� cos [�� (𝑡 − 𝑡2)] − 𝐶16�� sin [�� (𝑡 − 𝑡2)] (51)

where 𝐶15 and 𝐶16 are two free variables. Substituting thecontinuity conditions 𝑥𝑀+𝑚 = 0 and ��𝑀+𝑚 = V𝑀+𝑚 when 𝑡 =𝑡2 into (50) and (51), then the two variables can be obtainedby the following.

𝐶15 = V𝑀+𝑚��𝐶16 = − (𝑥1 − 𝑥𝑚)

(52)

As a further result, the displacement 𝑥 of the bolts duringthe elastic oscillation relative to the initial position of thewhole process can be given by the following.

𝑥 = 𝑥2 − 𝑥𝑀+𝑚 (53)

In particular, the second peak in the load-time curve isdue to the face pallet hitting the impact rod during the elasticrecovery. In the impact tensile test for the double CRLD boltsacting in parallel, the bullet hits the impact rod and the firstpeak is formed on the face pallet that is in close contact withthe impact rod. Subsequently, the CRLD bolts start to moveand the impact rod remains approximately stationary. Afterthe deformation of the bolts reaches the maximum value, thebolts recover elastically in the opposite direction. The secondpeak is generated when pallet hits the impact rod locatedin initial position again. The intensity of the second pulseis determined by the internal forces of the shanks when thedeformation of the bolts reaches the maximum.

Under the elastic impact load, the bolts only deformelastically and the sleeves and shanks are always relativelystationary. When the deformation of the bolts is maximum,the internal forces in the shanks of the double bolts, that is,the intensity of the second pulse, can be given by

𝑃𝑚𝑎𝑥2 = 2𝑘 ⋅max (𝑥) (54)

where 𝑃𝑚𝑎𝑥2 is the intensity of the second pulse loadand max(𝑥) denotes the maximum deformation of the boltsduring the whole process.

When the pallet recovers and reaches the initial position,the speed of the bolts attains a minimum value which is


Table 2: Experimental and calculated data under three gas pressures to determine the transmission coefficients.

Gas pressure (MPa) Bullet velocity (m/s) Shock wave in impact rod Shock wave in face pallet RatiosDuration (ms) Intensity (KN) Duration (ms) Intensity (KN) K1 K2

1.5 10.97 0.31 980.96 0.45 505.58 1.45 0.521.8 12.60 0.31 1126.70 0.45 610.89 1.45 0.542.0 14.48 0.31 1294.80 0.45 740.69 1.45 0.57

a negative value indicating the maximum speed in reverse.Therefore, the starting time 𝑇 of the second pulse satisfies

𝑥 (𝑇) = min (��) (55)

where 𝑇 is starting time of the second pulse load andmin(��) denotes the minimum speed of the bolts during thewhole process.

Under the stable impact load and unstable impact load,the bolts produce structural deformation, and the sleevesslide relative to the shanks.When the deformation of the boltsis maximum, the deformation of the shank is 𝑥𝑐𝑟 − 𝑥𝑚; hencethe intensity of the second pulse load is as follows.

𝑃𝑚𝑎𝑥2 = 2𝑘 (𝑥𝑐𝑟 − 𝑥𝑚) (56)

Different from being under the elastic impact load, whenthe speed of the bolts reaches aminimumvalue, the bolts onlycomplete the recovery of elastic deformation and the boltsdo not return to the initial position due to the presence ofstructural deformation. At the subsequent times, the palletcontinues to move in the maximum speed in reverse untilit collides with the impact rod resulting in the second pulseload. The starting time 𝑇 of the second pulse is

𝑇 = 𝑇1 + 𝑇2 (57)

where 𝑇1 is the time when the speed of the bolts reachesthe minimum value, and satisfies 𝑥(𝑇1) = min(��). 𝑇2 is thetime from the bolts speed reaching the minimum value to thepallet hitting the impact rod and satisfies 𝑇2 = −[max(𝑥) −𝑥(𝑇1)]/min(��).4. Verification of the Theoretical Results withImpact Tensile Test

In the impact tensile test for the double CRLD bolts actingin parallel, the bullet hits the impact rod and the shock waveis generated in the impact rod. According to the traditionalSHPB experimental theory [26, 27], the duration Δ𝑡∗ of theshock wave in the impact rod can be calculated by

Δ𝑡∗ = 2𝑙𝑐 (58)

where 𝑙 is the length of the bullet equal to 800mm and c isthe speed of the elastic stress wave in the impact rod equal to5190m/s. The intensity 𝑃∗

𝑚𝑎𝑥of the shock wave in the impact

rod can be given by

𝑃∗𝑚𝑎𝑥

= 12𝜌𝑐Vs (59)

where V is the striking velocity of the bullet equal to17.94m/s, 𝜌 is the density of the impact rod equal to7800 kg/𝑚3, and s is their contact area (i.e., the same cross-sectional area of the bullet and the impact rod) and can beobtained by the following.

s = 𝜋(0.0752 )2 ≈ 0.00441786 (𝑚3) (60)

When the shock wave in the impact rod propagates to thecontact surface with the face pallet, a portion of the energy isprojected into the face pallet, so that the double CRLD boltsacting in parallel produce corresponding responses. Let 𝐾1represent the ratio of the duration Δ𝑡 of the shock wave inface pallet relative to Δ𝑡∗, and let 𝐾2 denote the ratio of theintensity 𝑃𝑚𝑎𝑥 of the shock wave in the face pallet relative to𝑃∗𝑚𝑎𝑥

; then we can have the following.

Δ𝑡 = 𝐾1 ⋅ Δ𝑡∗ (61)

𝑃𝑚𝑎𝑥 = 𝐾2 ⋅ 𝑃∗𝑚𝑎𝑥 (62)

In order to determine the ratios 𝐾1 and 𝐾2, the impacttensile tests for the double CRLD bolts acting in parallelat the gas pressures of 1.5MPa, 1.8MPa, and 2.0MPa wereperformed. Monitoring the striking velocities of the bulletat the three different gas pressures, then the durations andthe intensities of the shock waves in the impact rod canbe calculated by (58) and (59), respectively. Based on themeasured durations and intensities of the shock waves in theface pallet, the corresponding𝐾1 and𝐾2 can be obtained (thespecific data is shown in Table 2).

Taking the average values of 𝐾1 and 𝐾2 as their referencevalues, and then𝐾1 = 1.45 and𝐾2 = 0.54. From (58) and (61),the duration of the shock wave in face pallet at the 1.0MPa gaspressure can be obtained by the following.

Δ𝑡 = 1.45 × 2 ∗ 0.85190 ≈ 0.45 × 10−3 (s) = 0.45 (ms) (63)

The intensity of the shock wave in the pallet, from (59)and (62), can be obtained by the following.

𝑃𝑚𝑎𝑥 = 0.54 × 12 × 7800 × 5190 × 6.14 × 0.0044≈ 2.9649 × 105 (N) = 296.49 (KN)

(64)

Combined with the other physical parameters of thespecimens for CRLD bolts (see Table 3), the theoretical load-time curve for the double CRLDbolts acting in parallel underimpact load can be obtained, as shown in Figure 10. The


Table 3: Parameters used in drawing the analytical curve.

Parameter valueConstant resistance 𝑃𝑐𝑟 (KN) 184.17Corresponding displacement on constantresistance 𝑥𝑐𝑟 (mm) 2.3

Coefficient on intensity of impact load 𝑃𝑚 (KN) 148.245Coefficient on duration of impact load 𝜆 (1/ms) 13.96Mass of single bolt shank and cone𝑚 (Kg) 3.26Mass of double sleeve and pallet 𝑀 (Kg) 35.68Stiffness of the shank 𝑘 (KN/mm) 79.8Sleeve elastic constant 𝐼

𝑠 (N/m3) 1.003 × 1011

Cone geometrical constant 𝐼𝑐 (m3) 2.902 × 10−6

Static frictional coefficient𝑓 0.1007Dynamic frictional coefficient𝑓𝑑 0.0879

(0.30, 271.06)

(0.225, 296.49)

Experimental dataAnalytical data

(1.97, 140.81)

(2.07, 114.63)

1 2 3 4 5 6 7 8 9 10 110Time (ms)

−50

0

50

100

150

200

250

300

Load

(KN

)

Figure 10: Comparison of experimental data and analytical dataabout the load of the double CRLDbolts. (The gas pressure is 1.0MPaand the speed of the bullet is 6.14m/s.)

load reaches the maximum value 296.49KN at 0.225ms andattains the second peak value 114.63KN at 2.07ms. From thecomparison of the experimental data and the analytical dataabout the load of the double CRLD bolts, it can be seen thattheir change trends and the two peak values are basicallyconsistent, which indicate the correctness and reliability ofthe theoretically dynamic model.

5. Discussion about the Dynamic Responses ofthe Double CRLD Bolts

In order to investigate the mechanical responses of thedouble CRLD bolts acting in parallel under impact loadswith different intensities and an identical duration (Δt =2ms), it is quite necessary to determine the correspondingranges of the intensities of the three different types of impactload (the specific values as shown in Table 4). Specificallywe, respectively, choose one from each of the three types of

impact load for comparison and analysis, which possesses theintensities of 100KN, 250KN, and 800KN (see Figures 11, 12,and 13, respectively).

In particular, it has been known that the second pulsein the load-time curve of the CRLD bolts is caused bythe experimental equipment. It does not belong to themechanical properties of the CRLD bolts. Therefore, it is notconsidered for the second pulse in the following analysis.

Figure 11(a) shows the displacement-time curve and theload-time curve of the CRLD bolts under the impact loadwith the intensity 100KN and the duration 2ms. The max-imum displacement 1.06mm of the bolts does not reachthe maximum elastic displacement 2.3mm, so there is norelative sliding of the sleeves and the shanks, only the elastictension and compression of the shanks themselves during theentire impact process. Both the load and the displacementof the bolts eventually become zero when the bolts tendto be stable. Figure 11(b) indicates the load-displacementcurve of the CRLD bolts under the elastic impact load. Theload firstly rises and then decreases with the increase ofthe displacement. However after the displacement reachesthe maximum value, the load begins to reduce with thedecrease of the displacement. Furthermore, the time of thedisplacement reaching its peak significantly lags behind thetime of the load achieving its maximum.

Figure 12(a) illustrates the displacement-time curve andthe load-time curve of the CRLD bolts under the impactload with the intensity 250KN and the duration 2ms. Themaximum displacement 2.74mm of the CRLD bolts exceedsthe maximum elastic displacement 2.3mm, so the boltsproduce structural deformation caused by the relative slidingof the sleeves and shanks. Specifically the sleeves and shanksbegin to slide relative to each other at 1.31mswhen the internalforce of the shank reaches the constant resistance 184.17KNas well as with the corresponding displacement 2.3mm. Therelative sliding ends at 1.61ms, at which the bolts achievethe maximum displacement. The impact load still continuesuntil it becomes zero at 2ms. The relative sliding has stoppedbefore the end of the impact load, which indicates the impactbeing the stable impact load. In addition the displacementfinally stabilized at 1.06mm after the oscillation.

Figure 12(b) indicates the load-displacement curve of theCRLDbolts under the stable impact load.The load firstly risesand then decreases with the increase of the displacement.However after the displacement reaches the maximum value,the load begins to reduce with the decrease of the displace-ment. Furthermore, the time of the displacement reaching itspeak significantly lags behind the time of the load achievingits maximum.

Figure 13(a) shows the displacement-time curve and theload-time curve of the CRLD bolts under the impact loadwith the intensity 800KN and the duration 2ms. The max-imum displacement 31.67mm of the CRLD bolt exceeds themaximum elastic displacement 2.3mm, so the bolts alsoproduce structural deformation caused by the relative slidingof the sleeves and shanks. Specifically the sleeves and shanksstart to slide relative to each other at 0.82ms when theinternal force of the shank reaches the constant resistance184.17KN as well as with the corresponding displacement


Table 4: Critical ranges for three different types of impact load.

Type Load intensities (KN) CharacteristicElastic impact load 𝑃𝑚𝑎𝑥 ≤ 218.74 Only elastic deformations without structural deformationsStable impact load 218.75 ≤ 𝑃𝑚𝑎𝑥 ≤ 412.18 End of structural deformations earlier than end of impactUnstable impact load 𝑃𝑚𝑎𝑥 ≥ 412.19 End of structural deformations later than end of impact

Displacement-time curveLoad-time curve

Elastic impact load(1.54, 1.06)

Ending of impact load

0 mm

−1.0−0.5

0.00.51.0

Disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 220Time (ms)

−200−150−100−50050100150200250

Load

(KN

)Pmax ≤ 218.74 KN

(a)

Maximum displacement


Maximum load

Load-displacement curve

020406080

100

Load

(KN

)

0.750.500.250.00 1.00 1.25−0.50−0.75 −0.25−1.00Displacement (mm)

(b)

Figure 11: Curves for double CRLD bolts acting in parallel under impact load with intensity 100KN and duration 2ms: (a) displacement-timecurve and Load-time curve and (b) load-displacement curve.


Stable impact load

(1.61, 2.74)Ending of impact load

Starting of relative slidingEnding of relative sliding

1.06 mm

2 4 6 8 10 12 14 16 180 2220Time (ms)

−0.50.00.51.01.52.02.53.0

Disp

lace

men

t (m

m)

−1000100200300400

Load

(KN

)218.75 KN ≤ Pmax ≤ 412.18 KN

(a)


Ending of relative sliding with maximum

displacement


Starting of relative sliding

Maximum load

0.0 0.5 1.0 1.5 2.0 2.5 3.0−0.5Displacement (mm)

050

100150200250

Load

(KN

)

(b)

Figure 12: Curves for double CRLD bolts acting in parallel under impact load with intensity 250KN and duration 2ms: (a) displacement-timecurve and Load-time curve and (b) load-displacement curve.

2.3mm. Although the impact load is completed at 2ms, therelative sliding of the sleeves and shanks does not stop. Therelative slippage still continues until 3.06ms, at which thebolts reach themaximumdisplacement 31.67mm.The relativesliding still takes place after the end of the impact load, whichdemonstrates the impact being the unstable impact load. Inaddition the displacement finally stabilized at 28.87mm afterthe oscillation.

Figure 13(b) indicates the load-displacement curve of theCRLD bolts under the unstable impact load. The load firstrapidly rises and then slowly decreases with the increase ofthe displacement. Interestingly, the displacement continuesto increase after the ending of the impact load. Furthermore,the time of the displacement reaching its peak signifi-cantly lags behind the time of the load achieving its maxi-mum.


Load

(KN

)


Unstable impact load

(3.06, 31.67)



Ending of relative sliding28.87 mm

05

101520253035

Disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 220Time (ms)

0150300450600750900

Pmax ≥ 412.19 KN

(a)

0150300450600750900

Load

(KN

)


Ending of relative sliding with maximum displacement Ending of impact load

Maximum load


2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 340Displacement (mm)

(b)

Figure 13: Curves for double CRLD bolts acting in parallel under impact load with intensity 800KN and duration 2ms: (a) displacement-timecurve and Load-time curve; and (b) load-displacement curve.

6. Conclusion

Under the impact loads with different intensities, the situa-tions of force and motion for the double CRLD bolts actingin parallel have great difference, which results in the differentmechanical responses. (i) Under relatively small loading, theCRLD bolts deform elastically and the deformation finallyreturns to zero. (ii) Under the high impact load, including thestable impact load and unstable impact load, the CRLDbolts undergo structural deformation after the initial elasticdeformation. The deformation of the CRLD bolts eventuallystabilizes at a certain amount caused by the relative sliding ofthe sleeves and shanks. The essential difference between thestable impact load and unstable impact load is that, under thestable impact load, no structural deformation will occur afterthe impact load ends; under the unstable impact load, thestructural deformation will still occur after the impact loadends.

In addition, under the three different impact loads, thetime of the displacement reaching its peak significantly lagsbehind the time of the load achieving its maximum. Underthe elastic impact load and stable impact load, there is acontraction of displacement before the end of the load.However, under the unstable impact load, the displacementcontinues to increase during the entire loading process.

Based on the above discussion about the mechanicalresponse properties of theCRLDbolts under the impact loadswith different intensities, some guidance can be provided forthe safe and stable support of the roadway with differentburied depth and lithology.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Theresearch work is supported by the Special Funds from theNationalNatural Science Foundation of China [grant number

51574248]; the National Key Research and DevelopmentProgram [grant number 2016YFC0600901]; and the OpenFoundation of the State Key Laboratory for Geomechanicsand Deep Underground Engineering, Beijing [grant numberSKLGDUEK1728].

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