Multiple Impact of Beam-To-beam

International Journal of Impact Engineering 31 (2005) 185–219

Multiple impact of beam-to-beam

X. Teng, T. Wierzbicki*

Impact and Crashworthiness Laboratory, Department of Ocean Engineering, Massachusetts Institute of Technology,

Room 5-218, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Received 2 July 2002; accepted 27 August 2003

Abstract

Based on the rigid-plastic string model, the problem of a beam impacted sequentially by numerous beamswith a high velocity is investigated using the wave propagation approach. Attention is focused on theresponse of the stricken beam. The closed-form expressions for deflection and tensile strain are obtained inthe double impact case. In the multiple impact case, a general recursion formula for the tensile strain as afunction of the number of the striking beam is derived. By assuming the tensile necking failure mode, thecritical impact velocity to fracture the stricken beam is predicted with the impact number specified.Alternatively, with the impact velocity given, the critical impact number of the striking beam is determined.Asymptotic analyses for two limiting cases with infinite and infinitesimal time interval are performed.r 2003 Elsevier Ltd. All rights reserved.

Keywords: Multiple impact; Beam-string; High velocity; Failure

1. Introduction

In the report on the September 11th attack released by Federal Emergency ManagementAgency (FEMA) in May 2002, a detailed survey of the damage of the airplane to the outer facadeof the Twin Towers was made [1]. Based on photos and video clips, it was determined that 31–36exterior columns of the North Tower were destroyed over portions of a four-story range, Fig. 1.When a scaled outline of the Boeing 767 is superimposed on the damaged area, it becomes clearthat the gash in the facade of approximately 31 m in length is shorter than the wing span of47:6 m; Fig. 1. The wing tip was deflected by approximately 8 m by the exterior columns duringthe impact process. Therefore, both the exterior column and the airplane wing must haveundergone considerable plastic deformation before rupture. As shown in Fig. 2, the structural

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*Corresponding author. Tel.: +1-617-253-2104; fax: +1-617-253-1962.

E-mail address: [email protected] (T. Wierzbicki).

0734-743X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijimpeng.2003.08.006

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Nomenclature

2b width of the stricken beams2bi width of the striking beamsc stress wave speed for the stricken beamE0 kinetic energy acquired by the stricken beam at the first impactDE kinetic energy acquired by the stricken beam at the second impacth thickness of the stricken beamhi thickness of the striking beamsI transverse momentum provided by the striking beamsIs considered momentum transferred to the stricken beamDIs momentum loss in the first impactm mass per unit length of the stricken beamM0 rigid body massmi mass per unit length of the striking beamsn impact number%N plastic tensile force in the stricken beam

t timet0 time intervalv0 impact velocityvcr critical impact velocity to fracture the stricken beamvj transverse velocity due to the jth impactv�j transverse velocity immediately after the jth impact’vj transverse acceleration due to the jth impactDvj velocity increment due to the jth impactW plastic axial stretching energy of the stricken beamw transverse deflectionw0 deflection slope of the stricken beam.w transverse accelerationDw0

j slope increment due to the jth impactx axial coordinate for the stricken beam0 @

@x

� jump of a given quantity across the wave front: @

@t

a mass ratiob wave speed ratiogj normalized slope after the jth impactdc displacement of the impact areae tension strainemax maximum tensile strainef fracture strain for the stricken beamen tensile strain after the nth impact

X. Teng, T. Wierzbicki / International Journal of Impact Engineering 31 (2005) 185–219186

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Z dimensionless time intervaly dimensionless parameterxj the jth wave front locationr material density of the stricken beamri material density of the striking beams0 plastic flow stress of the stricken beamssi plastic flow stress of the striking beamt time parameter

Fig. 1. Superimposition of a scaled outline of the Boeing 767 on the damaged facade of the north face of the North

Tower.

Fig. 2. Structural component arrangement inside an airplane wing.

X. Teng, T. Wierzbicki / International Journal of Impact Engineering 31 (2005) 185–219 187

arrangement of a modern airplane wing is quite complicated, consisting of open section beams,ribs, and a skin reinforced by stringers. Hence, the exterior column must have been impactedmultiply by the structural components of the wing when the airplanes crashed into the TwinTowers.

The immediate objective of the present paper is to develop a mathematically tractable,computational model for a multiple impact event among plastically deforming and fracturingbeams, where the striking beams represent the structural components of the wing, and the strickenbeam an exterior column of the Twin Towers. At the same time, the analysis is rather general andcan be applied to any beams of solid section made of same or different materials.

According to the FEMA report, the travelling velocities of the airplanes were 210 m=s (NorthTower, WTC 1) and 264 m=s (South Tower, WTC 2), respectively, when the airplanes crashedinto the building [1]. These are an order of magnitude higher than the velocity attained by droptowers or horizontal sledge facilities. In fact, most of the fine works published over the years byJones and his coworkers dealing with mass impact on beams apply to the lower end of the velocityspectrum up to 20 m=s [2–7]. In such cases, the bending response is dominant, and the deflectionof the beam is smaller than the beam thickness. As the impact velocity of the projectile increases,axial stretching becomes comparable to bending response. A coupled bending-tension solutionwas developed by Yu and Stronge [8]. The solution indicates that when the deflection of the beamexceeds the beam thickness, axial stretching becomes dominant over bending response. All of theabove references deal with the impact problem of rigid mass-to-beam. The interactive impactproblem between two deformable beams was investigated by Yu and his coworkers [9,10], wherethe bending response was taken into account while the axial stretching was neglected. Therefore,their solution also apply to the lower end of the velocity spectrum.

For sufficiently high velocity impact, a beam can be simplified as a string with flexural rigidityneglected. The problem of high velocity mass impact on a plastic string was formulated and solvedby Mihailescu et al. [11] using a rigorous shock wave formulation and by Wierzbicki and HooFatt [12] using the momentum conservation principle. Both solutions were shown to be identicalfor an infinite beam. No experiments are available in the literature to validate the closed-formsolution, but recently performed numerical simulations confirm the correctness of the approach[13]. Meanwhile, there are an abundance of test results on projectile impact on plates. Thesolution for the problem of mass impact on a beam was extended by Wierzbicki and Hoo Fatt [14]to the case of a large circular plate. Good agreements of transient deflection profiles were obtainedwith the experiments by Calder and Goldsmith [15]. The theoretically predicted ballistic limitsagree within 10% to the test results by Calder and Goldsmith [15] and the recently publishedexperiments by B^rvik et al. [16]. Hence, the validity of the wave propagation approach has thusbeen firmly established.

Since the September 11th attack, the Impact and Crashworthiness Lab at MIT has beeninvolved in the study of the airplane wing cutting through the exterior columns of the TwinTowers. A series of reports/papers was published on the response and failure of two or multipleimpacts over a large range of beam dimensions, materials, and impact velocities. In Ref. [17], thewing was modeled as a rigid mass due to heavy fuel tanks while the exterior column as a plasticbeam. The critical impact velocity to cut through the columns was estimated. Then, the wavepropagation approach was extended to the single impact case of beam-to-beam [18], in whichboth the wing and the exterior column were modeled as two plastically deformable strings. The

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closed-form solution for the deflection profile and tensile strains were obtained. Five fracturescenarios were identified. Two beams could be of different materials and cross sections.Depending on the combination of various parameters, either of the beams will fail first. In thepresent paper, we made an assumption that the striking beams fail first upon impact while thestricken beam continues to deform. This assumption further restricts the applicability of oursolution, but still a class of beams satisfying the solution is very broad. In a more recent paper[19], we have treated numerically in the impact of rigid mass-to-beam. This paper quantifies therange of the problem parameters, such as mass ratios, relative impact velocities, etc. for whichthrough thickness shear failure or tensile failure occurs.

In the present paper, the wave propagation approach is first used to investigate the doubleimpact case, where three deforming and fracturing beams are involved. Then, the solution isextended to a general case where a beam is impacted sequentially by arbitrary number identicalbeams. The present paper applies to a restricted range of geometrical parameters and impactvelocities for which the beam indeed be treated as a string, and the deflections are larger than thethickness of the beam. Consequently, the bending phase of the response which is relevant for thedeflection less than one thickness is neglected. It is assumed that all of the striking beams fail byshear plugging immediately upon impact. Attention is focused on the response of the strickenbeam. Of interest is to determine the transient velocity, the deflection profile, the tensile strain,and the critical impact velocity to fracture the stricken beam.

2. Double impact

Consider a stationary plastic beam subjected to double normal impact by two identical plasticbeams moving with the velocity v0 and separated by the time interval t0; Fig. 3. The width andthickness of the beam are denoted by 2bi and hi for the striking beams, and 2b and h for thestricken beam. The subscript i represents the striking beam. The mechanical properties of thebeam are defined by the mass density ri and the average plastic flow stress si for the striking

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Fig. 3. Schematic of double impact of beam-to-beam.


beams, r and s0 for the stricken beam. From the above parameters, one can uniquely define themass density per unit length mi ¼ 2ribihi for the striking beams and m ¼ 2rbh for the strickenbeam, respectively. The impact configuration is shown in Fig. 4, which also defines the axialcoordinate x in the stricken beam. For convenience, the origin of the axial coordinate is defined atthe edge of the impact area. Due to symmetry of the impact response about the impact area, theexpressions for deflection and plastic strain will be given only for the positive side of the axialcoordinate in the following derivation.

The impact velocity is high enough so that the response of the beam is governed by local inertiaand wave propagation. Far-field boundary conditions are not involved for sufficiently longbeams. Due to high impact velocities, the beams are subjected to moderately large deflection sothat the bending resistance of the beam becomes insignificant. It is assumed that two inequalitiesbi=hb1 and b=hib1 are satisfied. Otherwise, the problem becomes three dimensional and can onlybe treated by means of numerical methods.

2.1. Range of validity

Even an elastic impact of three bodies poses a considerable mathematical problem. The same iseven more true for an inelastic impact with possible fracture. Many impact scenarios coulddevelop depending on geometry and materials of two beams as well as impact velocities. Sometypical impact scenarios are shown in Fig. 5. Both beams can continue deforming plasticallywithout fracture (Case a), or the striking beam fails in shear while the stricken beam is deformingwithout fracture (Case b). It is also possible that it is the stricken beam that fractures and theimpacting beam does not (Case c). The solution in this paper is restricted to the case when thestriking beam fails immediately upon impact whereas the stricken beam keeps deforming.

Now, the deforming beam for which the transient velocity and displacement fields weredetermined in the previous publication by the present authors [18] is subjected to the second

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Fig. 4. Impact configuration denoting geometrical and mechanical parameters for the stricken and striking beams.


impact. It is assumed that the second striking beam also fails in shear (Case d) and the history ofthe deformation of the stricken beam is the subject of the present paper.

The question arises if the sequence of deformation mode to be treated in the paper is possible inpractical situations. To prove this point, we are showing results of numerical simulation of rigidmass-to-beam impact that could generate three different deformation and failure modes, see Fig. 6[19]. For low impact velocities below 100 m=s no failure of the beam takes place. In theintermediate range of the impact velocity (100–180 m=s), the response is governed by tensiletearing failure. For the impact velocity higher than 180 m=s; an instantaneous shear failure isobserved. This latter case is considered in the present paper.

As an illustration, Fig. 7 shows the deformation and fracture mode of two beams impacted at600 m=s [20]. This result was obtained using ABAQUS/Explicit with the fracture criterion ofaccumulated plastic strain and a cut-off value of negative triaxiality.

Two situations may develop later. The remainder of the striking beams could stick to theimpacted beam and move with it. Alternatively, the broken piece will slide off the impact area andwill no longer interfere with the stricken beam. Both cases will be considered in this paper. Forsimplicity, the latter case will be taken as an example to present problem formulation in thefollowing sections.

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Fig. 5. Four possible impact scenarios.


2.2. Problem formulation

The problem after the second impact is formulated by following the procedure for the singleimpact case of beam-to-beam [18], and for rigid mass impact on a plastic beam [12]. However,different from the single impact and the rigid mass impact where the stricken beam is at rest upon

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Fig. 7. Deflection of the single impact of beam-to-beam at t ¼ 48 ms: The initial impact velocity is V0 ¼ 600 m=s:

Fig. 6. Residual velocity versus initial impact velocity for various mass ratios.


impact, in the double impact case, the second impact occurs at the stricken beam with thetransient fields of velocity and deflection. Hence, it is an entirely new initial-boundary valueproblem.

Consider the stricken beam after the second impact. The local equilibrium equation for thestricken beam in the deformed region behind the first wave front is given by

ð %Nw0Þ0 ¼ m’v; 0pxpx1; ð1Þ

where %N ¼ 2s0bh is the axial tensile force; x1 ¼ ct is the location of the first wave front; v is thetransverse velocity; w is the transverse deflection of the stricken beam; and the symbols prime anddot denote differentiation with respect to the spatial and temporal coordinates, x and t;respectively. Because s0 is assumed to be constant, so is %N; and the governing equation can bereduced to the familiar wave equation

c2w00 ¼ .w; ð2Þ

where .w ¼ ’v; and c is the wave speed, defined by

c ¼

ffiffiffiffi%N

m

r¼

ffiffiffiffiffis0

r

r: ð3Þ

Due to the discontinuity of the transverse velocity field, two shock waves are generated in thestricken beam upon the first and second impact, respectively. The shock waves propagate awayfrom the impact area with the constant plastic wave speed c: All plastic deformations of the beamtake place at the wave front, where the material element is instantly stretched, rotated, andimparted a transverse velocity. The deformed regions behind the wave front undergo rigid bodymotion. Different from the single impact case, a material element in the stricken beam is subjectedtwice to plastic deformations due to the first and second shock waves. Hence, two regions undergorigid body motion with different transverse velocities denoted by v1 and v2; respectively. Bothvelocities are constant in space and vary with time. A typical profile of the velocity field after thesecond impact for the stricken beam and the striking beams is shown in Fig. 8.

The deflection and the transverse velocity of the beam should satisfy the kinematic and dynamiccontinuity conditions at the wave front [12]:

’w� þ cw0� ¼ 0;

%Nw0� þ mc ’w� ¼ 0; ð4Þ

where the symbol � denotes a jump of a given quantity across the wave front. In detail, thekinematic and dynamic continuity conditions at the first wave front, x ¼ x1; require

ð0 � v1Þ þ cð0 � w0jx¼x�1Þ ¼ 0;

%Nð0 � w0jx¼x�1Þ þ mcð0 � v1Þ ¼ 0; ð5Þ

where zero initial conditions in front of the first wave front are used. Similarly, the continuityconditions at the second wave front, x ¼ x2 ¼ cðt � t0Þ; are specified as

ðv2 � v1Þ þ cðw0jx¼x�2� w0jx¼xþ2

Þ ¼ 0;

%Nðw0jx¼x�2� w0jx¼xþ2

Þ þ mcðv2 � v1Þ ¼ 0: ð6Þ

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The problem also satisfies the boundary condition at x ¼ 0

2 %Nw0 ¼ 2mbi ’v; ð7Þ

where only the mass of the stricken beam at the impact area contributes to the inertia force, whichcorresponds to the case where no parts of the striking beams remain in contact with the strickenbeam after the impact. Eq. (7) means that the transverse shear force resulting from the axialstretch is equilibrated by the inertia force. It is not necessary to introduce interaction forcebetween the stricken beam and the striking beams in the above equation, because no moremomentum transfers among the beams after fracture of the striking beams. Without difficulty,one can derive boundary conditions for the other case where the remainder of the striking beammoves with the stricken beam.

In terms of v1 and v2; the governing equation (1) can be rewritten as

ð %Nw0Þ0 ¼ m’v2; 0pxpx2;

ð %Nw0Þ0 ¼ m’v1; x1pxpx2: ð8Þ

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Fig. 8. Transient velocity field in three beams after the second impact.


In each deformed region, both the slope and transverse velocity of the beam are continuous.Based on the assumption of the uniform distribution of the transverse velocity in each deformedregion, integrating the above equations with respect to x in the respective regions yields

%Nw0jx¼x�2� %Nw0jx¼0 ¼ m’v2ðct � ct0Þ; 0pxpx2;

%Nw0jx¼xþ2� %Nw0jx¼x�1

¼ m’v1ct0; x1pxpx2:ð9Þ

Substituting the corresponding boundary and dynamic continuity conditions, and combining theabove equations together gives the governing equation for the stricken beam with respect to thetransverse velocities

½bi þ cðt � t0Þ�’v2 þ ct0 ’v1 þ cv2 ¼ 0: ð10Þ

Similar to the case of rigid mass impact and the single impact of beam-to-beam, the abovegoverning equation can also be directly obtained from the principle of linear momentumconservation by differentiating the following equation with respect to time t:

2m½bi þ cðt � t0Þ�v2 þ 2mct0v1 ¼ I ; ð11Þ

where I is a constant denoting the total momentum supplied by the striking beams during thedouble impact process.

Note that the momentum transferred from the striking beam is a little underestimated, sinceonly the momentum in the impact area is taken into account. When both stricken beams aresheared off, some momentum of both striking beams away from the impact area is transferred tothe stricken beam. Preliminary finite element simulations of the single impact of beam-to-beamindicate that the failure duration is of the order of 100 ms [20], and thus the underestimation of themomentum imparted by the striking beams would not be significant. Estimate can be given on theamount of momentum loss during the fracture process. Assuming the duration of the shearfracture process, ts; is

ts ¼hi

3v0; ð12Þ

where the coefficient 1/3 in the expression for the duration of fracture comes from the propertythat shear crack travels at a speed approximately equal three times higher than the impact velocity[21]. The neglected momentum in the first impact is approximately given by

DIs ¼ ri2bihi2tscðv0 � v�1 Þ; ð13Þ

where v�1 is the instantaneous velocity of the striking beam at the impact zone immediately uponimpact. The considered momentum transferred from the first striking beam to the stricken beam,Is; is

Is ¼ ri2bihi2bðv0 � v�1 Þ: ð14Þ

The ratio of the neglected momentum to the considered momentum is

DIs

Is¼

1

3

hi

b

c

v0: ð15Þ

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In order for the problem to be one dimensional rather than three dimensional, we have alreadyassumed that hi5b: Therefore, the term DIs can be indeed disregarded as long as the impactvelocity v0 is of the same order as the transverse plastic stress wave speed c:

2.3. Assumptions

It can be observed that from the global equilibrium and the conditions of the kinematic anddynamic continuity, only one equation is found for the two unknown velocities v1ðtÞ and v2ðtÞ:Therefore, an additional condition is needed to solve the problem.

Different from the first shock wave which propagates in the stricken beam with zero initialconditions, the second wave propagates in the already deformed region of the stricken beam. Asstated earlier, the slope w0jx¼xþ2

is generated when the first wave arrives at x ¼ x2: Since then, theslope of the beam at x ¼ x2 does not change with time until the second shock wave arrives there.Hence, Dw0

2 ¼ w0jx¼x�2� w0jx¼xþ2

is the increment of the slope at x ¼ x2 due to the second shockwave. The kinematic and dynamic continuity conditions at the second wave front, x ¼ x2; Eq. (9),can be recast in the incremental form

Dv2 þ cDw02 ¼ 0;

%NDw02 þ mcDv2 ¼ 0; ð16Þ

where Dv2 ¼ v2 � v1:Now, we introduce an assumption that the incremental parts, Dv2 and Dw0

2; satisfy thegoverning wave equation at 0pxpx2;

ð %NDw02Þ

0 ¼ mD’v2; 0pxpx2; ð17Þ

where D’v2 ¼ ’v2 � ’v1: Then, the governing wave equation, Eq. (8), with regard to the velocity v1

can be extended from the region between the two wave fronts, x2pxpx1; to the whole deformedregion, 0pxpx1;

ð %Nw01Þ

0 ¼ m’v1; 0pxpx1; ð18Þ

where the deflection w1 is generated due to the transverse velocity v1: The above equation indicatesthat the first shock wave will continue to propagate to the far field after the second impact as if thesecond impact never happened. In fact, both shock waves propagate with the common wave speedin the same direction. It is impossible that the second shock wave would interfere or overtake thefirst one, and vice versa. Also, note that the governing wave equation is a homogeneous partialdifferential equation such that the superposition of the solution is applicable.

Based on the assumption, there are two governing equations (17) and (18) with regard to Dv2

and v1; respectively, both of which satisfy the continuity conditions at the corresponding wavefront, Eqs. (5) and (16). As shown in the preceding section, if the transverse velocity satisfies thegoverning equation and the continuity conditions, the corresponding transverse momentum isconserved during the propagation process. As shown in Fig. 9, the transverse velocity field in thestricken beam after the second impact is decomposed into two parts. Each part represents a shockwave propagating with the wave speed, c; and the transverse momentum is conserved for eachpart. It is convenient to derive the transverse velocity at any time from the principle of momentumconservation.

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2.4. Velocity history

The solution to the velocity and deformation before the second impact is the same as the singleimpact case of beam-to-beam, which was presented in Ref. [18]. Immediately upon the firstimpact, the velocity is

v�1 ¼2mib

2mib þ 2mbi

v0 ¼ av0; ð19Þ

where the velocity with the superscript � represents the instantaneous velocity upon impact;a denotes the relative weight of the stricken beam and the striking beam at the impact area,defined by

a ¼mib

mib þ mbi

; 0oao1: ð20Þ

At a time before the second impact, the principle of linear momentum conservation gives

2mðb0 þ ctÞv1 ¼ 2mb0v�1 : ð21Þ

Note that there is no more momentum transfer between the striking beam and the stricken beamafter the impact. From the above equation, one obtains the velocity v1

v1ðtÞv0

¼a

1 þ Zt=t0; ð22Þ

where the dimensionless parameter Z is defined by

Z ¼ct0

bi

¼1 � b1 � a

t0

t; ð23Þ

where the parameter Z is related to the wave speed ratio, b; and the time parameter, t; both ofwhich were defined in the single impact case [18]

b ¼mici

mc þ mici

; ð24Þ

t ¼mbi þ mib

mc þ mici

: ð25Þ

By means of b and t; one can make a comparison between the double impact and the singleimpact. Just before the second impact, t ¼ t0; the velocity v1 becomes

v1ðt0Þv0

¼a

1 þ Z¼ y: ð26Þ

For convenience, a new dimensionless parameter y is introduced in the above equation.

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Fig. 9. Velocity field in the stricken beam after decomposition.


Now, solve the velocity field in the stricken beam after the second impact. Immediately uponthe second impact, the velocity v�2 can be obtained from the momentum conservation at theimpact area

ð2mib þ 2mbiÞv�2 ¼ 2mibv0 þ 2mbiv1ðt0Þ; ð27Þ

where the first term on the right-hand side represents the momentum provided by the secondstriking beam; the second term denotes the existing momentum of the stricken beam at the impactarea. Using the dimensionless parameters a and Z; it is convenient to write v�2 as

v�2v0

¼ a 1 þ1 � a1 þ Z

� �: ð28Þ

From the momentum conservation for the incremental part as shown in Fig. 9, one gets thevelocity increment Dv2ðtÞ

Dv2

Dv�2¼

m2bi

m½2bi þ 2cðt � t0Þ�¼

1

1 þ Zðt=t0 � 1Þ; ð29Þ

where Dv�2 is the initial value of the incremental part, given by

Dv�2 ¼ v�2 � v1ðt0Þ ¼ av0ð1 � yÞ: ð30Þ

As stated earlier, the variation of the velocity v1 with time is independent of the second impact, i.e.the expression for the velocity v1 after the second impact is the same as Eq. (22). The velocity v2 isobtained by adding v1 and Dv2 together

v2

v0¼ a

1

1 þ Zt=t0þ

1 � y1 þ Zðt=t0 � 1Þ

� �: ð31Þ

Plots of the velocity variation with time for specific values of a and Z are shown in Fig. 10.

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Fig. 10. Velocity variation with time for v1 and v2 in the double impact case.


2.5. Deflection and plastic strain

The transverse deflection of the beams is calculated by integrating the transverse velocity withrespect to time. For the stricken beam, a point located at a distance x from the impact areaacquires displacement only after time t ¼ x=c: The transverse velocity at this point changes fromv1 to v2 at t ¼ t0 þ x=c: Hence, the transverse deflection of the stricken beam after the secondimpact can be expressed as

w ¼R t0þx=c

x=cv1 dt þ

R t

t0þx=cv2 dt; 0pxpx2;

w ¼R t

x=cv1 dt; x2pxpx1:

ð32Þ

Substituting the expressions for the velocities, and integrating Eq. (32), one obtains the transversedeflection of the stricken beam,

w

bi

¼av0

cln

1 þ Zt=t0

1 þ x=bi

� �þ

av0

cð1 � yÞ ln

1 � Zþ Zt=t0

1 þ x=bi

� �; 0pxpx2;

w

bi

¼av0

cln

1 þ Zt=t0

1 þ x=bi

� �; x2pxpx1:

ð33Þ

The first term on the right-hand side of Eq. (33) represents the deflection due to the velocity v1;and the second term denotes the deflection due to the velocity increment Dv2: Plots ofinstantaneous deflection profiles of the stricken beam are shown in Fig. 11.

From the intermediately large deflection theory, the axial tensile strain in the stricken beam isgiven by

e ¼1

2

@w

@x

� �2

¼

1

2

av0

c

� 2

ð2 � yÞ2 1 þx

bi

� ��2

; 0pxpx2;

1

2

av0

c

� 2

1 þx

bi

� ��2

; x2pxpx1:

8>>><>>>:

ð34Þ

Plots of the tensile strain variation along the beam for different values of Z are shown in Fig. 12. Itcan be seen that there is a jump of the tensile strain at the second wave front, x ¼ x2: In the range

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Fig. 11. Deflection profiles of the stricken beam after the second impact and compared to the case of the single impact.


of x2pxpx1; the tensile strain is the same as that in the case of the single impact. The maximumtensile strain emax always takes place at the impact point, i.e. x ¼ 0;

emax ¼1

2

av0

c

� 2

ð2 � yÞ2: ð35Þ

When the time interval t0 goes infinity, Z-N; the maximum tensile strain approaches anasymptote

emax ¼ 22 1

2

av0

c

� 2

; Z-N; ð36Þ

which means that the double impact becomes two separate, single impact. When the time intervalt0 goes zero, the maximum tensile strain approaches another asymptotic value,

emax ¼1

2

av0

c

� 2

ð2 � aÞ2; Z-0: ð37Þ

2.6. Kinetic energy and plastic energy

Immediately upon impact, the stricken beam acquires the kinetic energy

E0 ¼ 12

2mbiðv�1 Þ2 ¼ mbiðav0Þ

2: ð38Þ

At a time, tot0; before the second impact, the kinetic energy of the stricken beam becomes

E1ðtÞ ¼1

2ð2mbi þ 2mctÞv2

1 ¼E0

1 þ ct=bi

; tot0: ð39Þ

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Fig. 12. Strain variation along the stricken beam for different values of Z:


Upon the second impact, the newly acquired kinetic energy is

DE ¼1

2ð2mbiÞ½ðv�2 Þ

2 � v21ðt0Þ� ¼ E0 1 þ 2

1 � a1 þ Z

þa2 � 2a

ð1 þ ZÞ2

� �: ð40Þ

At a time, t; after the second impact, the total kinetic energy of the stricken beam becomes

E2ðtÞ ¼1

2½2mbi þ 2mcðt � t0Þ�v2

2ðtÞ þ1

2ð2mct0Þv2

1ðtÞ

¼E03 � 2y

1 þ Zt=t0þ

ð1 � yÞ2

1 þ Zðt=t0 � 1Þ

� �: ð41Þ

The loss of the kinetic energy during the wave propagation is absorbed through plasticdeformation

W ðtÞ ¼E0 þ DE � E2

¼E0 1 þ ð1 � yÞ2 þ 21 � y1 þ Z

�3 � 2y

1 þ Zt=t0�

ð1 � yÞ2

1 þ Zðt=t0 � 1Þ

� �: ð42Þ

For a rigid-perfectly plastic string, the loss of the kinetic energy is completely converted to theaxial stretching energy. For a plastic beam, some of the loss of the kinetic energy is dissipated inthe form of plastic bending, besides plastic axial stretching.

2.7. Critical impact velocities

A beam under high velocity impact would fail by either tensile tearing or shear plugging. Here,it is assumed that the stricken beam fails by tensile tearing. A constant critical plastic strain, ef ; isused to predict fracture of the stricken beam, due to its simplicity.

By setting the maximum plastic strain equal to the fracture strain, one can estimate the criticalimpact velocity of the striking beam, vcr; to fracture the stricken beam,

vcr

c¼

ffiffiffiffiffiffi2ef

pað2 � yÞ

¼ffiffiffiffiffiffi2ef

p ð1 � aÞ þ ð1 � bÞt0=ta½ð2 � aÞð1 � aÞ þ 2ð1 � bÞt0=t�

: ð43Þ

The component of the bending strain is not taken into account in the estimation of the criticalimpact velocity in the above equation.

It can be shown that the critical velocity always decreases as the time interval increases, Fig. 13,no matter whether a > b or aob: When the time interval t0 goes infinity, the critical velocitybecomes half of that in the single impact:

vcr

c¼

1

2


pa

: ð44Þ

In the preceding, the critical impact velocity is determined by differentiating the deflection toobtain the strain. In the following, we will show an alternative way to determine the criticalimpact velocity.

Considering that the transverse velocity is a function of time, independent of the spatialcoordinate, the deflection slope of the beam can be calculated by differentiating equation (32) with

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respect to x;

@w

@x¼ �

1

cv1

x

c

� � v1 t0 þ

x

c

� þ v2 t0 þ

x

c

� h i; 0pxpx2; ð45Þ

where Leibniz’s differential rule has been used. Specifically, the slope at the impact area, i.e. atx ¼ 0; is given by

@w

@x

��x¼0

¼ �1

c½v1ð0Þ � v1ðt0Þ þ v2ðt0Þ� ¼ �

1

c½v�1 � v1ðt0Þ þ v�2 �: ð46Þ

Three instantaneous velocities, v�1 ; v1ðt0Þ; and v�2 ; are involved to determine the slope at x ¼ 0;which have been given in Eqs. (19), (26), and (28), respectively. Note that these velocities areexactly obtained from the linear momentum conservation without introducing any additionalassumptions, hence the slope at x ¼ 0 is exactly determined. Substituting the expressions for thesevelocities into Eq. (46), the same critical velocity as Eq. (43) can be obtained after similar algebra.

This method is more convenient, and thus will be used in the following cases.

2.8. Double impact by two identical rigid masses

Consider another case that both striking beams break immediately upon impact, the remainderof the striking beams sticks to the stricken beam at the impact area and moves with it, Fig. 14. Infact, this case is equivalent to the double impact event of a plastic beam by two identical rigidprojectiles with the mass M0 ¼ 2mib: The problem can be completely solved in the same procedureas described in the preceding sections. For simplicity, only the critical impact velocity to fracturethe stricken beam will be derived in the following.

The simple method based on Leibniz’s differential rule will be used, in which only threeinstantaneous velocities v�1 ; v1ðt0Þ; and v�2 ; are involved to determine the tensile strain at x ¼ 0;Eq. (46). These velocities can be exactly solved from the principle of momentum conservation.

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Fig. 13. Critical velocity variation with time interval in the double impact.


Different from the first case, the masses of the remainder of two striking beams or two rigidbodies contribute to the transverse momentum in the stricken beam during the whole wavepropagation process. After the first impact, the momentum conservation in the stricken beamgives

½2mðbi þ ctÞ þ 2mib�v1 ¼ ð2mbi þ 2mibÞv�1 : ð47Þ

From the above equation, one can express the velocity v1ðtÞ in terms of the mass ratio, a; and thetime parameter, Z;

v1ðtÞv0

¼a

1 þ ð1 � aÞZt=t0: ð48Þ

Particularly, the velocity v1 at t ¼ t0 is given by

v1ðt0Þv0

¼a

1 þ ð1 � aÞZ: ð49Þ

Upon the second impact at t ¼ t0; the momentum conservation can be expressed as

ð4mib þ 2mbiÞv�2 ¼ 2mibv0 þ ð2mbi þ 2mibÞv1ðt0Þ; ð50Þ

which yields the velocity v�2v�2v0

¼a

1 þ a1 þ

1

1 þ ð1 � aÞZ

� �: ð51Þ

Substituting the expressions for v�1 ; v1ðt0Þ; and v�2 into Eq. (46), one can determine the slope andfurther the tensile strain at x ¼ 0:

emax ¼1

2

@w

@x

� �2

¼1

2

v0

c

� 2 a1 þ a

� �2

2 þ a�a

1 þ ð1 � aÞZ

� �2

: ð52Þ

Plots of the maximum tensile strain versus the mass ratio a at various values of Z are shown inFig. 15. It appears that the maximum tensile strain does not increase monotonically with the massratio for the time parameter Z in some range, which is different from the single impact event of a

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Fig. 14. Schematic of deformation and fracture of the beams in the double impact case.


plastic beam by a rigid mass. The condition under which the extreme value of the maximumtensile strain exists can be found by setting the differential of the maximum tensile strain withrespect to the mass ratio equal to zero,

@emax

@a¼ 0; ð53Þ

which after arrangement gives

Z2a4 � 2Za3 � ðZ2 þ 2ZÞa2 � ð2Z2 þ 2ZÞaþ 2ðZþ 1Þ2 ¼ 0: ð54Þ

With the value of Z given, the above polynomial equation in a can be solved numerically, Fig. 16.The root of the equation determines the location of the extreme value in Fig. 15. If Z ¼ 1:0; theextreme value of the maximum tensile strain occurs at a ¼ 1:0: If Zo1; there is no extreme valuesof emax; and the tensile strain always increases monotonically with the mass ratio. The extremevalue usually occurs at aE0:90 under the condition of Z > 1: In the case of the single rigid massimpact, a heavy impacting body is always easy to break the stricken beam, as shown byWierzbicki and Hoo Fatt [12]. However, in the case of the double rigid mass impact, there is anoptimal mass ratio as a function of the time interval, with which the stricken beam is the easiest tofracture.

Setting emax ¼ ef in Eq. (52), one can determine the critical impact velocity to fracture thestricken beam,

vcr

c¼


p 1 þ aa

2 þ a�a

1 þ ð1 � aÞZ

� ��1

; ð55Þ

Plots of the critical impact velocity versus the time interval are shown in Fig. 17.

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Fig. 15. Maximum tensile strains versus mass ratios for different values of Z in the case of double rigid mass impact.


2.9. Double impact by two intact striking beams

In the preceding sections, the cases that both striking beams undergo fracture immediately uponimpact were solved. Now, consider the third case that both striking beams deform plastically and neverfracture during the impact process. The striking beams will always contact with the stricken beam.

Assume that the maximum slope in the stricken beam occurs at x ¼ 0 after the second impact.The expression for the slope at the impact point is the same as Eq. (46), while the expressions fortwo instantaneous velocities, v1ðt0Þ and v�2 ; will be different.

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Fig. 17. Critical velocity variation with the time interval for different values of a and b:

Fig. 16. Critical mass ratio a determining the extreme value of the maximum tensile strain.


From the analysis of the single impact of beam-to-beam [18], the velocity v1ðt0Þ is given by

v1ðt0Þv0

¼ bþa� b

1 þ t0=t: ð56Þ

The transverse momentum is conserved at the impact area immediately upon the second impact,i.e. t ¼ t0;

2mibv0 þ ð2mib þ 2mbiÞv1ðt0Þ ¼ ð4mib þ 2mbiÞ v�2 ; ð57Þ

which gives the velocity v�2

v�2v0

¼a

1 þ aþ

1

1 þ av1ðt0Þ

v0: ð58Þ

Substituting the expressions for v�1 ; v1ðt0Þ and v�2 into Eq. (46), and setting emax ¼ ef ; one canobtain the critical velocity vcr to break the stricken beam in the case of both striking beamswithout fracture

vcr

c¼


p 1 þ aa

1 þ t0=t2 þ ð2 þ a� bÞt0=t

: ð59Þ

Plots of the critical velocity variation with the time interval are shown in Fig. 18. There are threecases depending on relative values of a and b: For the case of a ¼ b; the critical velocity is constantand independent of the time interval. For the case of a > b; the critical velocity decreases withthe time interval, and vice versa. If the time interval t0 goes infinity, the critical velocity

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Fig. 18. Comparison of the critical velocity to fracture the stricken beam among three cases: (Case 1) both striking

beams fracture upon impact and no parts of the striking beams remain in contact with the stricken beam; (Case 2) both

striking beams fracture upon impact and the remainder of the striking beam at the contact area moves with the striking

beam, or this case can be thought as double impact by a rigid mass; (Case 3) both striking beams keep intact during

impact process.


approaches an asymptote,

vcr

c¼


p 1 þ aa

1

2 þ a� b: ð60Þ

Fig. 18 illustrates comparison of the critical velocity to fracture the stricken beam among the threecases. It can be seen that the critical impact velocities always decrease with the time interval forthe three cases. The stricken beam is the most prone to fracture in the case where both strikingbeams break immediately upon impact and no parts remain in contact with the stricken beam.

3. Multiple impact

3.1. Problem formulation

Consider a stationary beam subjected to multiple impact by n identical beams moving with thevelocity, v0; the time interval, t0; and the total spacing, l ¼ ðn � 1Þv0t0; Fig. 19. Because totallyn þ 1 beams are involved during the impact process, there are many possible fracture scenariosleading to complicated transverse velocity field for every beam. Here, two simple cases will beinvestigated. The first case is that all of the striking beams break immediately upon impact; and noparts of the striking beams still rest on the stricken beam after the impact (Case 1). The second isthat all of the striking beams break immediately upon impact; and the remainder of the strikingbeams moves with the stricken beam, as shown in Fig. 14 (Case 2). Case 2 can also be consideredas the multiple impact by identical rigid bodies with the mass M0 ¼ 2mib: In both cases, attentionis focused on the response of the stricken beam. For simplicity, Case 1 will be taken as an exampleto present the problem formulation and the solution procedure.

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Fig. 19. Schematic of multiple impact of beam-to-beam.


After a new impact, two new shock waves are generated at the impact area and propagates tothe far field along both sides of the stricken beam. An additional plastic deformation in thestricken beam is instantaneously generated at the new wave front. The region behind the newwave front is subjected to rigid body motion again. Hence, similar to the case of the doubleimpact, the velocity and the acceleration between two wave fronts are constant in space and varywith time. A typical transverse velocity field in the stricken beam after the nth impact is shown inFig. 20.

Following the procedure for the double impact event, it can also be shown that from the globalequilibrium, the kinematic and dynamic continuity conditions, there is only one governingequation for the problem of the multiple impact, which is related to n unknown velocities after thenth impact. Hence, the problem cannot be solved without introducing additional conditions orassumptions.

Prior to this, let us look at the kinematic and dynamic continuity conditions at the jth wavefront, which are given by

ðvj � vj�1Þ þ c ðw0jx¼x�j� w0jx¼xþj

Þ ¼ 0;

%Nðw0jx¼x�j� w0jx¼xþj

Þ þ mcðvj � vj�1Þ ¼ 0; ð61Þ

where xj ¼ bi þ c ðt � jt0Þ is the location of the jth wave front. Both conditions can be recast in theincremental form

Dvj þ cðDw0jÞ ¼ 0;

%NðDw0jÞ þ mcDvj ¼ 0; ð62Þ

where Dvj ¼ vj � vj�1 is the velocity increment; and Dw0j is the slope jump at the jth wave front.

Eq. (62) indicates that the increment of the plastic slope is proportional to the velocity increment.Similar to the case of the double impact, it is assumed that the velocity increment Dvj and thecorresponding plastic slope increment Dw0

j satisfy the governing wave equation:

ð %NDw0jÞ0 ¼ mD’vj; xj�1pxpxj: ð63Þ

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Fig. 20. Schematic of the velocity field in the multiple impact case.


It can be shown that the transverse momentum is conserved with regard to the velocity incrementDvj; if the governing equation, the kinematic and dynamic continuity conditions are satisfied. Thevelocity field in Fig. 20 can be decomposed into n parts, Fig. 21. Every part represents a shockwave propagating with the same wave speed c: The transverse momentum for every part isconserved during the shock wave propagation process. Thus, the problem becomes amenable.

3.2. Velocity history

To develop the solution procedure, assume that the velocity field after the jth impact has beendetermined. Then the solution process for the next impact corresponding to the ð j þ 1Þth impact istypical and can be applied repetitively until the nth impact is completed.

At t ¼ jt0; the stricken beam is impacted by the ð j þ 1Þth striking beam. The transversemomentum is conserved at the impact area upon impact, given by

mi2bv0 þ m2bivjð jt0Þ ¼ ðmi2b þ m2biÞv�jþ1; ð64Þ

where the subscript j represents the jth impact; v�jþ1 is the instantaneous velocity at the impact areajust upon the ð j þ 1Þth impact; and vjð jt0Þ is the velocity of the stricken beam at t ¼ jt0 after thejth impact. From the above equation, it is convenient to express v�jþ1; in terms of the mass ratio, as

v�jþ1 ¼ av0 þ ð1 � aÞvjð jt0Þ: ð65Þ

The transverse velocity increment due to the ð j þ 1Þth impact is defined by

Dvjþ1ðtÞ ¼ vjþ1ðtÞ � vjðtÞ; tXjt0: ð66Þ

As assumed earlier, the transverse momentum is conserved for this velocity increment as theð j þ 1Þth shock wave propagates away from the impact area. Hence, Dvjþ1 at a time, t; is given by

Dvjþ1ðtÞDv�jþ1ð jt0Þ

¼m2b0

m2b0 þ m2cðt � jt0Þ¼

1

1 þ Zðt=t0 � jÞ; tXjt0; ð67Þ

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Fig. 21. Decomposition of the velocity field in the multiple impact case.


where Dv�jþ1 is the velocity increment immediately upon impact, i.e. at t ¼ jt0;

Dv�jþ1 ¼ v�jþ1 � vjð jt0Þ ¼ a½v0 � vjð jt0Þ�: ð68Þ

The total velocity vjþ1 at the region near the impact area is a summation of all of the velocityincrements:

vjþ1ðtÞ ¼ vjðtÞ þ Dvjþ1ðtÞ ¼Xjþ1

k¼1

DvkðtÞ; ð69Þ

where Dv1ðtÞ ¼ v1ðtÞ: Particularly, at t ¼ jt0; the velocity vjð jt0Þ is given by

vjð jt0Þ ¼Xj

k¼1

Dvkð jt0Þ

¼Dv�1

1 þ jZþ

Dv�21 þ ð j � 1ÞZ

þ?þDv�j

1 þ Z; ð70Þ

where Eq. (67) has been used.Based on the above solution process, it is easy to develop a calculation routine to determine the

velocity of the stricken beam at any time. Plots of the velocity versus time for specific values of aand Z are shown in Fig. 22.

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Fig. 22. Temporal variation of the velocity of the stricken beam in the multiple impact event.


3.3. Deflection

The deflection of the stricken beam at a distance x from the impact area after the nth impact isgiven by

w ¼Z t0þx=c

x=c

v1 dt þZ 2t0þx=c

t0þx=c

v2 dt þ?þZ t

ðn�1Þt0þx=c

vn dt; 0pxpc½t � ðn � 1Þt0�: ð71Þ

Similar expressions can be given for the deflection in other regions. Specially, the displacement atthe impact area, i.e. at x ¼ 0; is

dc ¼Z t

0

Dv1 dt þZ t

t0

Dv2 dt þ?þZ t

ðn�1Þt0

Dvn dt: ð72Þ

Fig. 23 shows comparison of the displacement history at the impact area among three cases.

3.4. Tensile strain and critical impact velocity

In this section, the simple method based on Leibniz’s differential rule will be used to derive thetensile strain at the impact area and further the critical impact velocity to break the stricken beam.From Eq. (71), differentiating the deflection with the spatial coordinate x; the slope of the beam atx ¼ 0 after the nth impact is given by

@w

@x

��t¼ðn�1Þt0

¼ �1

c½v1ð0Þ � v1ðt0Þ þ v2ðt0Þ � v2ð2t0Þ þ?þ vnððn � 1Þt0Þ�

¼ �1

c½v�1 � v1ðt0Þ þ v�2 � v2ð2t0Þ þ?þ v�n �: ð73Þ

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Fig. 23. Comparison of the displacement at x ¼ 0 among three cases.


From Eq. (68), the above equation can be rewritten as

@w

@x


¼ �1

c½Dv�1 þ Dv�2 þ?þ Dv�n � ¼ �

1

c

Xn

j¼1

Dv�j ; ð74Þ

where Dv�1 ¼ v�1 : The above equation indicates that the plastic slope of the stricken beam at x ¼ 0is proportional to the summation of all of the velocity increments upon impact. This equation willbe used to derive a recursion formula for the plastic slope later on.

Substituting the expression, Eq. (68), for every term in the right-hand side of the aboveequation, one gets

@w

@x


¼ �nav0

cþ

ac½v1ðt0Þ þ v2ð2t0Þ þ?þ vn�1ððn � 1Þt0Þ�: ð75Þ

From Eq. (70), the above equation can be expanded as

@w

@x


¼ � nav0

cþ

ac

Dv�11 þ Z

� ��

þDv�1

1 þ 2Zþ

Dv�21 þ Z

� �þ?þ

Dv�11 þ ðn � 1ÞZ

þDv�2

1 þ ðn � 2ÞZþ?þ

Dv�n�1

1 þ Z

� ��:

ð76Þ

Further, collecting common terms in the above equation, one obtains

@w

@x


¼ � nav0

cþ

a1 þ Z

1

cðDv�1 þ Dv�2 þ?þ Dv�n�1Þ

þa

1 þ 2Z1

cðDv�1 þ Dv�2 þ?þ Dv�n�2Þ þ?þ

a1 þ ðn � 1ÞZ

1

cðDv�1 Þ: ð77Þ

As indicated in Eq. (74), the above equation can be rearranged as

@w

@x


¼ � nav0

cþ

a1 þ Z

@w

@x


þa

1 þ 2Z@w

@x


þ?þa

1 þ ðn � 1ÞZ@w

@x

��t¼0

: ð78Þ

For simplicity, introduce a dimensionless parameter gj to denote the slope at x ¼ 0 after the jthimpact,

gj ¼ �@w

@x

��t¼ð j�1Þt0

c

av0: ð79Þ

In terms of the definition of gj; Eq. (78) becomes a recursion formula for the plastic slope

gn ¼ n �a

1 þ Zgn�1 �

a1 þ 2Z

gn�2 �?�a

1 þ ðn � 1ÞZg1; ð80Þ

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where g1 ¼ 1: For the case of Z > 2; an approximate closed-form solution for gn can be obtained byneglecting high order terms with regard to Z

gn ¼ n � aXn�1

j¼1

n � j

1 þ jZ: ð81Þ

A short computational routine is developed to calculate the plastic slope with the dimensionlessparameters given.

In terms of the dimensionless slope gn; the tensile strain in the stricken beam after the nth impactis given by

en ¼1

2

av0

c

� 2

g2n: ð82Þ

It is interesting to discuss two limiting cases: Z-N and 0, which are helpful to understand thegeneral case derived in the preceding. If the time interval Z goes infinity, the transverse velocity inthe stricken beam becomes infinitesimal until the next impact. That is, the velocity vjð jt0Þ inEq. (75) can be neglected. The tensile strain approaches an asymptote

en ¼1

2

av0

c

� 2

n2; Z-N: ð83Þ

On the other hand, if the time interval becomes infinitesimal, no shock waves have sufficient timeto propagate away from the impact area until the whole impact process is finished. In this case, themomentum conservation in the impact area after the jth impact is expressed as

2mibv0 þ 2mbiv�j ¼ ð2mib þ 2mbiÞv�jþ1; ð84Þ

where the velocity in the impact area before the next impact is still v�j : The velocity v�jþ1 is given by

v�jþ1 ¼ av0 þ ð1 � aÞv�j : ð85Þ

A recursion formula with respect to the difference in the velocities can be obtained from the aboveequation,

v�jþ1 � v�j ¼ ð1 � aÞðv�j � v�j�1Þ: ð86Þ

This recursion formula gives the closed-form solution for v�jv�jv0

¼ 1 � ð1 � aÞj: ð87Þ

Note that in this case,

Dv�j ¼ v�j � v�j�1: ð88Þ

Substituting the above expression into Eq. (74), the slope of the stricken beam after the nth impactdepends only on the velocity upon the last impact,

@w

@x


¼ �1

c½v�1 þ v�2 � v�1 þ?þ v�n � v�n�1� ¼ �

v�nc: ð89Þ

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Hence, the tensile strain becomes

en ¼1

2

v0

c

� 2

½1 � ð1 � aÞn�2; Z-0: ð90Þ

As shown in Figs. 24 and 25, the final tensile strain increases with the mass ratio and the timeinterval.

Similar to the double impact case, it is assumed that the stricken beam fails by tensile tearing,and the stricken beam does not have any damage before the final impact. Hence, the criticalvelocity to fracture the stricken beam is given by setting en ¼ ef in Eq. (82),

vcr

c¼


pagn

: ð91Þ

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Fig. 25. Tensile strain versus time intervals for different mass ratios for Case 1.

Fig. 24. Tensile strains versus mass ratios for different time intervals for Case 1.


Alternatively, with the impact velocity v0 given, one can determine the critical impact number ncr

to fracture the stricken beam, as shown in Fig. 26. The discrete points are calculated at specificvalues of a and Z: These points are fitted with two smooth solid curves for two cases.

3.5. Multiple rigid mass impact

In the preceding section, the case that no parts of the striking beams remain in contact withthe stricken beam after the impact was formulated and solved. Here, it is assumed that all of thestriking beams break immediately upon impact, and the remainder of the striking beams at thecontact area sticks to the stricken beam and moves with it. Similarly, this case is equivalent tothe case of the stricken beam impacted multiply by the identical rigid masses with the massM0 ¼ 2mib:

The problem is solved in the same procedure as that in the preceding case. Assume that we havesolved the velocity field after the jth impact. Now, we want to determine the velocity after theð j þ 1Þth impact. The momentum conservation upon the ð j þ 1Þth impact at the impact area isexpressed as

2mibv0 þ ð j2mib þ 2mbiÞvjð jt0Þ ¼ ½ð j þ 1Þ2mib þ 2mbi�v�jþ1; ð92Þ

which gives the instantaneous velocity, v�jþ1;

v�jþ1

v0¼

ajaþ 1

þ 1 �a

jaþ 1

� �vjð jt0Þ

v0: ð93Þ

Based on the assumption, the transverse momentum is conserved at any time with regard to thevelocity increment Dvjþ1:

½2mbi þ ð j þ 1Þ2mib�Dv�jþ1 ¼ ½2mbi þ ð j þ 1Þ2mib þ 2mcðt � jt0Þ�Dvjþ1ðtÞ; ð94Þ

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Fig. 26. Tensile strain versus impact number.


where Dv�jþ1 is the velocity increment upon the ð j þ 1Þth impact, defined by

Dv�jþ1

v0¼

v�jþ1

v0�

vjð jt0Þv0

¼a

jaþ 11 �

vjð jt0Þv0

� �: ð95Þ

In terms of a and Z; Dvjþ1ðtÞ can be expressed as

Dvjþ1ðtÞDv�jþ1

¼jaþ 1

jaþ 1 þ ð1 � aÞZðt=t0 � jÞ: ð96Þ

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Fig. 28. Comparison of tensile strains versus time interval between Case 1 and Case 2.

Fig. 27. Comparison of tensile strains versus mass ratio between Case 1 and Case 2.


Particularly, at t ¼ ð j þ 1Þt0;

Dvjþ1ðð j þ 1Þt0ÞDv�jþ1

¼jaþ 1

jaþ 1 þ ð1 � aÞZ: ð97Þ

With every velocity increment known, one can obtain the velocity vjðtÞ from Eq. (69), and furtherthe slope at x ¼ 0 can be obtained from Eq. (74). Different from the preceding case, no recursionformula was found. A short computational routine was developed to calculate the tensile strainwith the mass ratio, a; the time parameter, Z; and the impact number, n; given. Numerical resultsfor specific cases are shown in Figs. 27 and 28, and compared to Case 1. It appears that the tensilestrains in Case 1 are much larger than those in Case 2. Similar to the double impact case, thetensile strain does not increase monotonically with the mass ratio, Fig. 27.

Similar to the preceding case, two limiting cases are investigated as follows. First, if the timeinterval t0 goes infinity, the velocity in the stricken beam becomes infinitesimal until the nextimpact. Hence, in this case,

Dv�j ¼ v�j : ð98Þ

The total mass at the contact area in the stricken beam increases by 2mib after every impact. Afterthe jth impact, the velocity v�j is obtained from the momentum conservation,

Dv�jv0

¼v�jv0

¼2mib

j2mib þ 2mbi

¼a

ð j � 1Þaþ 1: ð99Þ

Substituting the expression for every velocity into Eq. (74), one can obtain the slope at x ¼ 0; andfurther the tensile strain in the stricken beam after the nth impact

en ¼1

2

v0

c

� 2 Xn

j¼1

a1 þ ð j � 1Þa

" #2

; Z-N: ð100Þ

As shown in Fig. 28, the tensile strain approaches to an asymptotic value as the time intervalincreases.

Now, consider another limiting case that the time interval t0 becomes infinitesimal. No shockwaves propagate away from the contact area until the whole impact process is finished. Similar tothe limiting case with Z-0 in Case 1, the slope after the nth impact depends only on the velocityupon the last impact, v�n : v�n can be obtained from the momentum conservation

v�nv0

¼n 2mib

n 2mib þ 2mbi

¼na

ðn � 1Þ aþ 1: ð101Þ

Hence, the tensile strain en after the nth impact is given by

en ¼1

2

v0

c

� 2

¼na

ðn � 1Þ aþ 1

� �2

; Z-0: ð102Þ

Fig. 29 shows the relations between the tensile strain and the mass ratio for different values of Z:The critical mass ratio a corresponding to the extreme value of the tensile strain increases with thetime interval Z; and approaches unity as Z becomes infinity.

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Note that no assumptions are introduced in the asymptotic analysis. Good agreements betweenthe asymptotic analysis and the general solutions for large values of Z verify the correctness of theadditional assumption introduced in Section 3.1.

4. Conclusions

The problem of multiple impact of beam-to-beam was formulated based on the rigid-plasticbeam-string model. In the case of the double impact, the closed-form solutions were obtained forthe deflection profile, the transverse velocity, and the tensile strain. By assuming the failure modeof tensile necking, the critical impact velocity to fracture the stricken beam was predicted. In thecase of the multiple impact, a recursion formula with respect to the impact number was found tocalculate the tensile strain. A computational routine was composed to solve the critical impactvelocity to fracture the stricken beam with the impact number given, and to determine the criticalimpact number with the impact velocity given. Asymptotic analyses were performed for twolimiting cases that the time interval goes either infinity or is kept infinitesimal.

To the best of the authors’ knowledge, no experiments are available in the literature to validatethe closed-form solution in the present paper. Numerical simulations will be performed toinvestigate the multiple impact problem using finite element methods.

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Fig. 29. Tensile strains versus mass ratios for different values of Z for Case 2.


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