Collision between a ring and a beam

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International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

Collision between a ring and a beam H.H. Ruan, T.X. Yu∗ Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 4 June 2003; received in revised form 4 September 2003; accepted 27 November 2003

Abstract With car–parapet collision accidents in mind, a normal collision between a free-0ying half ring and a simply supported beam with/without axial constraints is studied, in which an elastic–plastic half ring with an attached mass and the elastic–plastic beam are taken as the simplest models of a car and a parapet, respectively. Particular attention is paid to the energy partitioning between the two structures and the evolution of the contact regions during collision. A mass–spring 7nite di8erence (MS–FD) model is employed whilst the large de0ection and axial stretching/compression are incorporated. The numerical results show that the less sti8 (i.e. softer) structure will dissipate more energy and the contact regions will move away from the initial contact points. With the increase of the relative thickness of the beam to the ring, the 7nal deformation of the half ring will transform from a “U” shape to a “W” shape. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Ring; Ring-on-beam collision; Collision between two structures; Mass–spring 7nite di8erence (MS–FD) mode

1. Introduction Collisions between two structures occur in various engineering scenarios in our daily lives. In most cases, both of the colliding structures are deformable and absorb energy. A typical case is the collision between a vehicle and a roadside parapet—a frequently occurring traBc accident along highways. In order to better understand this kind of phenomena, leading to a better design of the roadside safety systems, the fundamental features of the deformation of the two colliding structures must be studied. However, most of the previous works on impact mechanics focused on a single structure’s dynamic response to an impulsive force or to a rigid body impact. These works neglected the in0uence of the relative sti8ness of the striker (the vehicle) and the target (the parapet) on the energy partitioning. If all the kinetic energy of the vehicle is assumed to be merely absorbed by the ∗

Corresponding author. Tel.: +852-2358-8652; fax: +852-2358-1543. E-mail addresses: [email protected] (H.H. Ruan), [email protected] (T.X. Yu).

0020-7403/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2003.09.025

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Nomenclature b C c DR D d E F G g h KR K0 L l M m N R t V X; Y Yb Yr yb yr        Y !

width of rectangular cross-section elastic constant of angular spring non-dimensional elastic constant of the angular spring restoring energy plastic energy dissipation non-dimensional plastic energy dissipation Young’s modulus force mass of a half of rigid block non-dimensional mass of the half of rigid block height of the rectangular cross-section remaining kinetic energy initial kinetic energy carried by a half ring length of a segment non-dimensional length of a segment bending moment non-dimensional nodal mass axial force radius of a half ring response time nodal velocity nodal coordinate displacement of mid-point of beam (Fig. 12(a)) displacement of the rigid plate pressing on a half ring (Fig. 17(a)) non-dimensional displacement of mid-point of beam non-dimensional displacement of rigid plate apply on a half ring non-dimensional tensile modulus of a bar non-dimensional mass per unit length strain non-dimensional bending moment non-dimensional axial force angle between two connected bars mass per unit length yield stress of material non-dimensional response time inclined angle of a bar

()

d()=d

•

Subscripts 0 b

initial state beam

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c p T r x y local softer sti8er

1753

contact fully plastic tensile half ring x direction y direction local contact softer (i.e. less sti8) structure sti8er structure

Superscripts in ex ◦

internal external last iteration step

parapet, the designed guardrail system would be too sti8. This design methodology would make the situation more dangerous, since the relatively less sti8 (i.e. softer) vehicle would have to dissipate more energy, according to our previous analysis on the collision between two deformable structures [1,2]. In order to propose a simple theoretical model concerning the coupled deformation of the two colliding structures, the collision between a free 0ying half ring and a simply supported beam with or without axial constraints is studied, in which the elastic–plastic half ring with an attached mass and the elastic–plastic beam are taken as the simplest models of a car and a parapet, respectively. Particular attention is paid to the energy partitioning between the two structures and the variation in the contact regions during collision. The analytical studies on the dynamic response of structures under impact have been carried out over 50 years. Since the pioneering works were accomplished by Lee and Symons [3] and Parkes [4], the rigid, perfectly plastic material model has been most widely adopted to study the dynamic behavior of structures subjected to intense dynamic loading. By neglecting the elasticity and strain-hardening of the material, this idealization signi7cantly simpli7es the deformation mechanism of the structure without losing the key features of its dynamic response. Under this idealization the plastic deformations are localized in the structures, and for 1-D structures, such as beams or rings, the localized plastic regions can be regarded as plastic hinges. With the variation of loading condition and geometry con7guration, the plastic hinge may change its position, i.e., a “traveling hinge”. The concept and formulation of traveling hinge were 7rst introduced by Lee and Symons [3] and then widely adopted in many analytical solutions, which include not only the dynamic response of single structures subjected to dynamic loadings [5,6] but also the collision between two beams [1,2]. This kind of solution was named a complete solution since it satis7ed all kinetic and dynamic conditions, e.g., the equations of motion, the initial and boundary conditions, while it does not violate the yield criterion at any point within the structure.

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Fig. 1. A MS–FD discretization of a cantilever beam.

Although very successful, it should be noted that almost all the complete solutions were obtained only for very simple structures, e.g., a straight beam, or for very simple loading conditions, e.g., a step loading. The small de0ection assumption was always necessary. The increase in the complication of geometry and loading conditions give rise to diBculties in the selection of an appropriate deformation mechanism among a number of alternatives, and in the formulation of the equations of motion. Questions, like how many hinges are enough or where the traveling hinges emerge, may obstruct one’s e8ort in solving a complicated dynamic problem. This is the reason why no complete solution has been obtained for the dynamic response of a ring subjected to an impact although many e8orts have been made. In addition, the incorporation of other factors such as the large deformation and elastic e8ects make the complete solution less attainable. Although the 7nite element method (FEM) can always be used as a powerful tool for any speci7c dynamic problem, it is too expensive to reveal the e8ects of various parameters on the 7nal result, e.g., energy partitioning or 7nal deformation. Hence, the semi-analytical mass–spring 7nite di8erence (MS–FD) model is employed in this paper. The MS–FD model discretizes a 1-D structure into some rigid elements connected by 0exible joints, as shown in Fig. 1, at which the elastic–plastic angular springs represent the elastic–plastic behavior of the joints while the mass of the structure is lumped at the nodes. Obviously, the MS–FD model has much lower degrees of freedom than the 7nite element method for the same level of discretization. And this phenomenological model is convenient for a semi-analytical approach because it naturally brings in non-dimensional variables that represent the system. The MS–FD method was 7rst proposed by Hou et al. [7], aiming to study the elastic e8ect on the dynamic response of a cantilever beam and to compare with the complete solution and the 7nite element simulation. Very good agreement between the MS–FD result and FE simulation are reported in [7] where only 20 elements were used in the MS–FD simulation of a cantilever beam. Owing to the ease of implementation, this model has also been adopted to study the quasi-static large elastic–plastic deformation of 0exural beams [8]. In this paper, the MS–FD model will be reformulated to incorporate large deformation and axial force, while local contact spring [2] will be employed to simulate the contact between two structures.

2. Problem description As shown in Fig. 2, consider a free-0ying half ring of uniform rectangular cross-section and radius R moving in a horizontal plane with initial downward velocity V0 . A block of mass 2G is attached on the ring to model the undeformable portions of a vehicle, such as the engine and the rear parts. Assume that at time t = 0, the ring normally strikes the mid-point of a simply supported beam of uniform rectangle cross-section and length 2Ls . Due to symmetry, only a half of each structure is considered. The global coordinate system is de7ned in Fig. 2, where Y is the transverse direction of the beam.

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X R

Vo Y

Ls

Ls

Fig. 2. A half ring with an attached rigid mass block impinging onto a simply supported beam with/without axial constraint.

To facilitate our analysis, the geometrical and material parameters and the variables of motion are normalized (refer to [5]) as
M r R  L X Y = ; = ; l= ; x= ; y= ; v=V ; Mpr r R R R Mpr
• Mpr NR FR = ; f= ; =t ; ( ) =d()=d ; (1) 3 Mpr Mpr r R where M denotes the bending moment and the subscript p represents the fully plastic value,  denotes the mass per unit length of the beam or the ring, X and Y are the nodal coordinates, V denotes nodal velocity, L represents the length of a segment, N is the axial force, while F represents any other force, t is the response time counted from the beginning of collision. The subscripts r pertain to the ring. 3. The mass–spring nite dierence (MS–FD) model The MS–FD discretization of the present problem is sketched in Fig. 3. As illustrated in Ref. [7], the angular springs around the nodes connect the neighboring bars and simulate the elastic–plastic bending behavior of the original structure. They are characterized by elastic constant C and yield moment Mp . The constitutive relation of these springs relates the bending moment and the change in relative angle S as  o   + cS; if |o + cS| ¡ p ;   if o + cS ¿ p ; (2)  = p ;    − ; o if  + cS 6 −  ; p

p

where c = C=Mpr and p = Mp =Mpr are the non-dimensional elastic constant and yield moment, respectively, o is the bending moment in the last iteration step. Strictly speaking, this constitutive relation is valid for an ideal sandwich Euler–Bernoulli beam, but generally it could serve as a useful approximation for any solid cross-sections.

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X The contact springs Y

Fig. 3. MS–FD discretization of the ring-on-beam collision.

In the real applications, since the guardrail system can be treated as an in7nitely long beam mounted on many poles, the segment between two poles can be regarded as an axially constrained beam. Thus the massless bars that connect two nodes are herein not rigid as proposed in Ref. [7] since the axial deformation is unavoidable. The dynamic response of the target beam in the present paper will be analyzed under the conditions with or without axial constraint for comparison. The constitutive relation in the axial direction is also assumed to be elastic, perfectly plastic, as formulated by  o   + ST ; |o + ST | ¡ p ;   NR o + ST ¿ p ; = (3) = p ; Mpr    − ; o  + S 6 −  ; p

T

p

where  = ET R=Mpr and T = Sl=l, in which ET is the tensile modulus and  is the non-dimensional counterpart, Sl is the length change and positive under tension, ST is the change of the tensile strain, o is the axial force in the last iteration step. In the present MS–FD discretization, since the axial force and the bending moment are decoupled as the characteristic of the 0exible joint and the bar, respectively, an independent yield criterion has to be employed:          = 1; or    = 1: (4)  p   p  The yield surface is then a square shown in Fig. 4. 3.1. The equations of motion The MS–FD model is composed of joints and bars. The lumped masses are attached at the joints, so that they represent the inertia e8ect of the structure; since the bars are massless the forces acting on each bar should be in equilibrium.

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M/Mp 1

-1

1

N/Np

-1

Fig. 4. The independent yield criterion of the bending moment and the axial force.

f jy

I

f

j

in jy

I

f jinx

f jx

f Ixex

x

f kiny

γΚ

k

f jx f jy

y (a)

y

j

f kinx

f Iex y

x

ωj

γΙ

K

(b)

Fig. 5. Free-body diagram of MS–FD elements in the global coordinate system: (a) a node, (b) a bar.

At any joint (or node), a shear force and an axial force are applied, resulting from the two bars connected to it. Fig 5(a) sketches a free body diagram of a single node I of mass mI . Two elastic, in in in perfectly plastic bars, i.e., bar j and bar k, are connected to node I. fjinx , fjy , fkx and fky denote the ex ex forces acting on node I along the x and y direction, resulting from the bars. fIx and fIy denote the external forces acting at node I. The bold subscripts I, j and k pertain to node I, bar j and bar k, respectively. In the following, the bold uppercase and lowercase subscripts represent the numbering of nodes and bars, respectively. The superscripts “in” and “ex” denote the internal force resulting from a bar and the external force acting at a node, respectively. Thus, the equations of motion of node I are written as ex fIx + finx − fjinx = mI xTI ;

ex in in fIy + fky − fjy = mI yT I ;

(5a,b)

where xI and yI denote the position coordinates of node I, and xTI and yT I are the corresponding accelerations. It is noted that in the MS–FD model, the external forces are only acting at the nodes. Hence, if the structure is subjected to a distributed force, the equivalent nodal forces should be determined by lumping the force on the neighboring bars. The bars are assumed to merely deform axially, hence their displacement and deformation are fully controlled by the displacements of the two end nodes. Fig. 5(b) sketches the free body diagram of

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bar j. The two end nodes are distinguished as the starting node I and the ending node K so as to describe the direction of this bar. The positive directions of the forces acting on the bar as well as the corresponding coordinate system are shown in the Fig. 5(b). Since the bar is assumed massless, the moment equilibrium is given by K − I + lj (fjy cos !j − fj x sin !j ) = 0;

(6)

where !j is the inclined angle of bar j in the global coordinate system; the non-dimensional bending moment I and K are calculated from Eq. (2) according to the change of relative angle of the connected bars in the current step. Suppose joint (node) I connects two bars i and j, the relative angle of bar j to bar i is then given as I = ! j − ! i :

(7)

The axial equilibrium gives fj x cos !j + fjy sin !j = j :

(8)

Herein, the axial force j is calculated from Eq. (3) by noting that the elongation of bar j is attributed to the position change of nodes I and K. in Noting that fj x and fjinx , and fjy and fjy are two pairs of action and reaction force, Eqs. (6) and (7) can be recast into in cos !j − fjinx sin !j ); K − I = lj (fjy

(9a)

in sin !j + j = 0: fjinx cos !j + fjy

(9b)

in The internal forces of bar j , fjinx and fjy , are then solved as −   K I fjinx = −j cos !j − sin !j ; lj  K − I in fjy = −j sin !j + cos !j : lj

(10a) (10b)

It is noted that the bending moment and axial force as well as the rotation angle of the bar in the right-hand side of Eq. (10a, b) can be calculated from the deformation of the structure, which are known in each iteration step when the Runge–Kutta method is used. Furthermore, the equations of motion for the whole system are completely decoupled in each element as Eqs. (5a, b) and (10a, b), unlike the large and complicated system-rigidity matrix in the 7nite element formulation. Hence, the dynamic solution by using the MS–FD model will be much easier and less expensive than the 7nite element method. 3.2. Discretization of the ring and the beam For any kind of discretization, the mass at a nodal point, the parameters of the angular spring at a 0exible joint and the tension/compression rigidity of a bar should represent the mechanical property of a certain portion of original structure in the corresponding region. As shown in Fig. 6(a), the segment ABC is discretized by two bars i and j of length li and lj , respectively, which are connected

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A i D

K B

K B

j

i

E

C C

(a)

A (b)

Fig. 6. The discretization of structures: (a) for the elements not on boundary, (b) for an element on boundary.

at node K. D and E are the mid-points of the segments AB and BC, respectively. Thus, the mass mK is assumed equal to the mass of segment DBE of uniform density, i.e., 1 (li + lj ): (11) 2 While, the elastic capacity of the angular spring at 0exible joint K is assumed equal to the elastic capacity of the segment DBE of length (li + lj )=2. Herein the sandwiched 1-D structure is still adopted to simplify the real structure, so that the equivalence of elastic capacity are obtained as mK =

Mp2 (li + lj ) Mp2 ; = 2CK 2EI 2

(12)

where E is the elastic modulus of the structure and I is the secondary moment of the cross-section. Thus, the non-dimensional elastic constant of the angular spring is given as cK =

CK 2EI = : Mpr Mpr (li + lj )

(13)

The normalized fully plastic bending moment of the joint (or the spring) is still p as that of the original structure. Herein the subscript index of the full plastic bending moment at joint K is omitted since the structures (the ring and the beam) are assumed uniform. The bars i and j have the same tension and compression behavior as segment AB and BC in the original structure, hence the non-dimensional tensile modulus is =

EbhR ; Mpr

(14)

where b and h are the width and height of the rectangular cross-section. The yield stress in tension and compression is p as the original structure. If the elements are on the boundary as shown in Fig. 6(b), only a half side is considered. Thus Eqs. (11) and (13) are recast as mK =

1 li ; 2

(15)

cK =

CK 4EI = : Mpr Mpr li

(16)

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S O

C

E

Fig. 7. The MS–FD discretization of a half ring and a beam.

In the present study on the collision between a half ring and a beam, one-quarter of the ring is uniformly divided into 15 segments while a half of the beam is divided into 25 segments, as shown in Fig. 7. Numerical studies showed that further re7ned discretization does not notably a8ect the result. Node G, as shown in Fig. 7, is assigned to represent the mass attached on the half ring, so its mass is given as mG = g;

(17)

where g = G=r R with G being the half of the mass of the rigid block attached to the half ring. The discretization in the beam is re7ned in the possible contact region OC of length 2R, as shown in Fig. 7, where the beam is uniformly divided into 20 segments. Herein, the possible contact region is de7ned arbitrarily to ensure that the contact occurs inside this region. The whole quarter of the ring is de7ned as the contact region in the present analysis. This de7nition provides the set of nodes for searching the contact pairs and for calculating the contact deformation and contact forces. Obviously a smaller but more precise de7nition of the contact region will shorten the contact searching period and save the computational time. In the MS–FD model, the contact is also assumed to occur between nodes. The localized contact behavior is characterized by the massless contact springs as sketched in Fig. 3 and discussed in Ref. [2]. For simpli7cation, in the present analysis the direction of the contact springs attached to the two contacting nodes is assumed to be collinear with the connection line of the two contacting nodes, that is, only the normal contact is considered while the friction is neglected. The two contacting nodes are de7ned when the distance between them is smaller than a critical value. The mechanical properties of the contact springs are hard to quantify. Ruan and Yu [2] proposed and compared several local contact models. In the MS–FD model, di8erent local contact models may greatly a8ect the contact searching and calculation algorithm. Especially when the interaction between two neighboring nodes needs to be considered, the complexity will dramatically increase. However, it has been revealed that as the energy partitioning is concerned, the local contact always dissipates very small portion of the initial kinetic energy and it does not notably a8ect the global energy partitioning between two colliding structures, especially for the structures of solid cross-section [2]. Hence, the contact spring can be modeled in the simplest manner, which mainly serves as a medium that relates the interaction force and the approach of the two structures. As shown in Fig. 8, the contact between node I in the ring and node J in the beam are simpli7ed as the uniaxial compression of the two 7ctitious blocks ABCD and EFGH. The Poisson e8ect and the shear deformation are neglected, which

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

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Fig. 8. A local contact model.

indicates the decoupled contact behavior between two neighboring nodes in a structure. Resembling the de7nitions of the equivalent mass and the elastic capacity of angular spring attached at the node, block ABCD and EFGH represent part of the original structures, in which A, B, E and F are mid-points of the original segments that are represented by bars i, j, k and l. Noting that the notable deformation occurs in the region close to the contacting surfaces and vanishes in the region close to the free surfaces, we roughly presume that only halves of the thickness in contact structures are subjected to uniaxial compression, as sketched in Fig. 8. Thus the mechanical properties of the contact spring can be easily obtained from the contact blocks. For instance, the elastic, perfectly plastic constitutive relation of the contact spring associated with node I, which is characterized by block ABCD (Fig. 8), is de7ned as  (li + lj ) (li + lj ) Sh (li + lj ) Sh Sh   Fco + Eb ; if 0 ¡ Fco + Eb ¡ Yb and co + ¿ cp ;   2 h=2 2 h=2 2 h=2     (li + lj ) (li + lj ) Sh (li + lj ) ; if Fco + Eb ¿ Y b ; Fc = Y b (18)  2 2 h=2 2      Sh   0; ¡ cp ; if co + h=2 where Fc is the contact force, which can be treated as the external force in Eq. (5a, b). Its non-dimensional counterpart is given by fc = Fc R=Mpr . Fco is the contact force in the last iteration step, Sh is the change of the compression length in the current iteration step, Y is the yield stress of the material, co is the total strain of the contact spring in the last step, while cp is the plastic compressive strain. Fig. 9 sketches the relation of these variables and the constitutive law of the contact spring. 3.3. Boundary and initial conditions The beam, aiming to model the roadside parapet, is assumed to be simply supported. Hence, the boundary node in the beam’s MS–FD model, represented by node E (Fig. 7), satis7es the restrain that yE = 0:

(19)

If the beam is axially constrained, one more restrain is added as xE = 0:

(20)

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ε cp

εc

εc Fig. 9. The relationship between the contact force and the approach of the two contact parts.

Since only halves of the half ring and the beam are considered, the symmetrical condition requires xS = 0;

xO = 0;

(21)

where S and O are the nodes on the symmetry axis in the half ring and the beam, respectively (Fig. 7). Since the mass block attached on the ring is assumed to be rigid, it renders xG = 0:

(22)

Moreover, the simple support condition implies the free rotation of the boundary segment of the beam, so that the angular spring adjacent to the boundary o8ers no resistance, that is, cE = 0:

(23)

Initially, the ring freely 0ies along the y-direction, so that the initial condition is given as yI = v0 ; xI = 0;

at

= 0; I ∈ the nodes in the ring:

(24)

4. Numerical simulations and discussions 4.1. The e8ect of relative sti8ness on the energy partitioning In engineering designs, the thickness of structures is usually an alterable parameter rather than other dimensions. Hence the relative thickness should be taken as an important factor that a8ects the energy partitioning between the two colliding structures. The energy-partitioning diagrams a8ected by the relative thickness for the beam-on-beam collision were discussed in Refs. [1,2], showing that the less sti8 structure always dissipates more energy.

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1 0.9

Energy partitioning

0.8 0.7

Beam Ring Local contact Total plastic dissipation

0.6 0.5 0.4 0.3 0.2 0.1 0 0

(a)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Thickness ratio hb /hr 1 0.9

Energy partitioning

0.8 0.7

Beam Ring Local contact Total plastic dissipation

0.6 0.5 0.4 0.3 0.2 0.1 0 0

(b)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Thickness ratio hb /h r

Fig. 10. The e8ect of the thickness ratio on energy partitioning between the ring and the beam: (a) for the simply supported beam without axial constraint, (b) for the simply supported beam with axial constraint.

For the present problem, suppose that both the beam and the ring are made of the same materials. The material and geometrical parameters as well as the initial velocity are taken as E = 70 GPa; Y = 140 MPa;  = 2680 Kg=m3 br = bb ; R=br = 10; R=hr = 10; Lb = 3R; V0 = 30 m=s; v0 = 0:82; g = 10: Figs. 10(a) and (b) illustrate the variation of energy partitioning with the thickness ratio for two cases that (a) the beam is simply supported without axial constraint; and (b) the beam is simply supported with axial constraint. Again, it is noted that with the increase of the thickness ratio hb =hr , the beam dissipates more energy, indicating that less sti8 structure bears more plastic dissipation. It is also noted that the local contact deformation only dissipates a very small portion of the initial kinetic energy. For an axially constrained beam with the attached mass g = 10, Fig. 11 depicts the energy partitioning pattern by showing the dissipations due to bending deformations and axial deformations. It is noted that no energy is dissipated by the axial deformation of the ring, indicating that the plastic deformation mechanism of the ring is dominated by bending, even when very large deformation occurs. The plastic dissipation due to axial deformation of the beam monotonically decreases with

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Beam's bending deformation Ring's bending deformation Beam's axial defoemation Ring's axial deformation

0.9

Energy partitioning

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Thickness ratio hb/hr

Fig. 11. Partitioning of the plastic energies dissipated by bending and axial deformations, as functions of the thickness ratio.

20 16

yr

12

mp

p

mp

8

np

np

4

lp

0

P (a)

0

(b)

0.04

0.08

0.12

0.16

0.2

yb/lb

Fig. 12. The load-carrying capacity of an axial constrained beam: (a) a simply supported, axially constrained beam subjected to central loading, (b) the load–displacement curve resulted from the rigid, perfectly plastic material idealization.

the increase of the thickness ratio, while the dissipation due to the bending deformation of the beam keeps increasing when the thickness ratio varies from 0 to 3.2. This phenomenon can be understood by noting the following two factors: (1) In the range of 0 –3.2 of the thickness ratio, the plastic deformation in the axial direction is an indication of large deformation. As shown in Fig. 12(a), consider a simply supported rigid, perfectly plastic beam with axial constraint subjected to a force at its mid-point, by using the independent yield criterion, the quasi-static load displacement relation can be formed as  mp l2b − yb2 yb p = 2 np  ; (25) + l l2b + yb2 l2 + yb2 and sketched in Fig. 12(b) for the case of np = 40, mp = 1, and lb = 3. It should be noted that as the load carrying capacity during large deformation is concerned, the contribution of the fully plastic axial tensile force increases with the increase of deformation, while the contribution of

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the fully plastic bending moment decreases. This indicates that when the thickness ratio ranges from 0 to 3.2, the deformation level of the beam is controlled by the axial sti8ness. (2) The fully plastic axial force is proportional to the thickness, while the fully plastic bending moment is proportional to the square of the thickness. Fig. 11 indicates that the increase of the beam’s fully plastic bending moment with the thickness results in the increase of the dissipation by plastic bending, because the deformation is controlled by the magnitude of the fully plastic axial force rather than the fully plastic bending moment. In the range of 3.5 – 4 of the thickness ratio, the plastic dissipation due to axial deformation vanishes, indicating that the deformation is dominated by the bending rigidity. It is observed that in this range, the larger is the fully plastic bending moment, the smaller is the dissipation in the beam. 4.2. Local contact and its dissipation Although the local contact deformation (i.e., the deformation of the contact springs) dissipates a very small portion of the total energy, it is still worthwhile to study the evolution of the contact region as well as the separation and multi-contact phenomenon, because they signi7cantly a8ect the global deformation of two colliding structures. Fig. 13(a–d) depicts the deformation of the ring and the beam in di8erent instants during the dynamic response for the case of g = 10 and hb =hr = 2:5 while the beam is under axial constraint. Fig. 13(e) demonstrates the evolution of the contact points, in which the nodes in the half ring and the beam are both numbered from the central point, that is, the index 0 represents the nodes at the mid-points of the half ring and of the beam; the larger is the index, the farther away from the mid-point is the node. Thus, Fig. 13(e) illustrates that the contact region moves away from the mid-points of the ring or the beam and 7nally, the half ring deforms into a “W” shape. Actually, with the increase of the sti8ness of the beam, the 7nal con7guration of the half ring changes from a “U” shape to a “W” shape, indicating that more and more energy is dissipated by the ring. Figs. 14(a) and (b) show the two cases in which a U-shaped and a slightly W-shaped 7nal con7guration of the half ring are obtained for the cases of g = 10, hb =hr = 1 and g = 10, hb =hr = 2, respectively. In the following discussion, it will be shown that the two di8erent 7nal con7guration shapes of the ring pertain to signi7cantly di8erent energy-partitioning patterns. The contact force varying with the response time is depicted in Fig. 15(a), where only the initial very brief period is demonstrated. It is noted that the contact force oscillates in a very high frequency. This high-frequency oscillation can be attributed to the 0exural wave e8ects. Since a 7nite time is needed for the propagation of bending deformation, the vibration of the local segments results in the oscillation of the contact force. The non-dimensional natural frequency of a typical segment as depicted in Fig. 15(b) is found in the order of 103 , which is in the same order of the oscillation frequency of the contact force shown in Fig. 15(a). It is observed that the contact force drops to zero, in some short intervals, indicating the separation of the two structures. Similar phenomenon is previously observed in Ref. [2]. The separation and multi-collision occur a few times in the collision process, while the real separation of the two structures occurs at a later time, as shown in Fig. 15(c), which depicts a 7ltered low frequency curve obtained by using a moving average method.

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Fig. 13. The deformation of the ring and the beam in di8erent instants after collision and the variation of the contact region for a beam with axial constraint, g = 10 and hb =hr = 2:5: (a) = 0:15; (b) = 0:49; (c) = 0:75; (d) = 1:50; (e) the variation of the contact nodes with the response time .

Fig. 14. The 7nal con7guration of the ring: (a) a “U” shape, (b) a slightly “W” shape.

The energy dissipated in the local contact region (i.e., the deformation of the contact springs) notably relies on the initial velocity, as demonstrated in Fig. 16, which is based on the thickness ratio hb =hr = 1 and the axially constrained beam. The horizontal axis is the initial kinetic energy, the variation of which results from varying the initial velocity and/or the attached mass. It is observed

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

1767

Fig. 15. The variation of the contact force with the response time : (a) in the initial short period, (b) a typical MS–FD segment, (c) in the whole dynamic response.

that with the increase of the initial kinetic energy, the local contact dissipation does not change if the initial velocity is 7xed, while it increases if the initial velocity of the ring increases. 4.3. Energy dissipation process 4.3.1. Energy dissipation in a quasi-static process For the case of a quasi-static ring-on-beam compression, in which the inertia e8ect is negligible, the energy partitioning between the ring and the beam is dictated only by the relative strength of the two structures. The strength of a structure herein refers to the structure’s static load-carrying capacity. For the present ring-on-beam collision case, since the contact region continuously evolves during deformation, it is hard to obtain the corresponding load–deformation behavior for the individual ring or beam. Thus, as a rough approximation, we employ the quasi-static behaviors of a half ring

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H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

Energy dissipated by contact springs

0.08

Varying velocity Varying mass

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0

5

10

15

20

25

Non-dimensional initial kinetic energy

Fig. 16. The e8ects of the attached mass and the initial velocity on the local energy dissipation.

P Yr

(a)

(b) Analytical solution MS-FD solution

20

p=PR/Mpr

16 12 8 4 0 0

(c)

0.1

0.2

0.3

0.4

0.5

yb=Yb/R

Fig. 17. The quasi-static load–deformation behavior of an end-constrained ring under a compression of a rigid-plate: (a) the sketch of an end-constrained half ring subjected to compression of a rigid-plate, (b) the side-constrained half ring in Ref. [9], (c) the load–deformation behavior of the half ring.

under the compression of a rigid plate (Fig. 17(a)) and a beam subjected to a load at its mid-point (Fig. 12(a)) to represent their load-carrying capacity during large deformation, while both structures are assumed to be rigid and perfectly plastic. Reddy and Reid [9] analytically obtained the quasi-static

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

1769

p 20

Ring 16

Simply supported beam without axial constrain

12 8

Initial kinetic energy

4

0 0.6

0.4

0.2

yb

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yr

Fig. 18. The load–deformation behavior of an end-constrained ring and a beam without axial constraint (hb =hr = 3).

behavior of a half ring under side constraint, as depicted in Fig. 17(b). Their problem is somewhat di8erent from the present one, but a similar analytical method can be adopted and the details are given in the Appendix. Fig. 17(c) sketches the analytically obtained load–deformation behavior of the half ring under compression, in which p is the non-dimensional load applied on the half ring, while yr is the non-dimensional displacement of the rigid plate. The ring’s load–deformation behavior can also be obtained by employing the current MS–FD model to solve the case that the half ring collides with a rigid ground at a very low velocity (0:05 mm=s for a ring of radius 50 mm). It can be seen that the analytical solution and the MS–FD solution agree well. As an example, by taking hb =hr =3 and a beam without axial constraint, Fig. 18 combines the static load–deformation behavior of the ring and the beam in a same diagram. In this kind of diagrams the areas under the curves represent the energy dissipated by the structures, while the summation of them should be equal to the initial kinetic energy. For the present case it is noted that the collapse load of the ring is higher than the load-carrying capacity of the beam so that the beam will dissipate all the initial kinetic energy. 4.3.2. Energy dissipation in a dynamic process However, as a result of the inertia e8ect in the dynamic collision process, the ring does dissipate a portion of the initial kinetic energy, as seen in Fig. 10(a) for the same case as the quasi-static one (i.e. hb =hr = 3). In order to elaborate the dynamic e8ect on the energy partitioning, we divide structural response into two stages according to our study in Ref. [10]: the very brief collision stage and the global deformation stage. Collision stage: Evidently, in an early stage of collision, the velocity di8erence exists between the ring and the beam, and the contact pair (i.e., the region in the vicinity of the mid-points) must experience velocity changes, so that the part with larger velocity is decelerated and the part with smaller velocity is accelerated. Hence an instant exists at which the contact pair 7rst attains a common velocity. The time period from the initial contact to that instant at which a common velocity is reached is named the collision stage. The inertia of the structure herein plays a role to prevent the contact pairs from sudden velocity change, and leads to the local contact deformation. It should be noted that unlike the quasi-static process, the contact force between the two colliding structures is now related to the local contact deformation, which could be much larger than the

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structure’s 0exural strength. Under the action of such a larger contact force, the collision stage is very brief and the energy is mainly dissipated by the local deformations rather than structures’ global deformations. Since the velocity discontinuity is severe at this stage, most of the local contact dissipation is accomplished within this stage. The importance of the collision stage lies not only on the local contact dissipation, but also on the local velocity changes, which produce the structural deformation in the later response. It is noted that the velocity change in the collision stage is a very localized behavior, which mainly occurs in the vicinity of the contact region. This is because the large contact force can form a plastic deformation zone around the contact region, while the 7nite structural 0exural strength (e.g., the plastic bending moment) is not able to render the corresponding velocity change in the other parts of the structure in such a short period. Since only the material within the vicinity of the contact region su8ers a notable velocity change, the magnitude of this change or the magnitude of the 7nal common velocity mainly depends on the mass of the local parts. Noting that the attached mass on the ring is allocated far from the local contact region, the energy dissipated by the local contact deformation thus will not be a8ected when the attached mass is varied, as shown in Fig. 16. Global deformation stage: The response after the collision stage is characterized by the structures’ global deformation and the associated energy dissipation. Hence, it is termed the global deformation stage. For each structure in the collision system the local velocity change can be regarded as an impulse applied in the vicinity of the contact region. If this impulse is small, it will act as a disturbance that propagates in the whole structure, and the structure will experience elastic vibration. However, if the impulse is large, the plastic zone/zones must appear and some of them may propagate (e.g., in the form of traveling hinges) and a portion of the kinetic energy will be dissipated. This explains why the ring dissipates energy for the dynamic collision case of hb =hr = 3 while it dissipates no energy in the quasi-static process. The plastic deformation in the ring ceases when the ring restores to its non-dissipation stage, in which the relative motion of particles in the ring is in elastic vibration only while the whole structure undergoes a rigid-body motion. The instant at which the sti8er structure transforms from an energy-dissipation state to a non-dissipation state is termed the restoring instant, and the corresponding energy dissipation in the sti8er structure is termed the restoring energy DR . It is noted that at the restoring instant, both the structures still carry some kinetic energy, termed the remaining kinetic energy K R . Although the multi-collision may happen in the later response, the velocity di8erence in the contact region will be much smaller compared with that in the collision stage. Hence, the velocity change (i.e., the impulse) in the contact region can only generate the elastic vibration in the sti8er structure, while the remaining kinetic energy will entirely goes to the less sti8 structure. Fig. 19(a) shows the energy dissipation process for the case of hb =hr = 3 and g = 10. It is observed that the local contact dissipation is accomplished almost immediately, indicating the briefness of the collision stage. After the ring (the sti8er structure in this case) ceases to dissipate energy at the restoring instant ( ∼ = 0:75) all the remaining kinetic energy will be dissipated by the beam (the less sti8 one in this case). It is evident that the above energy dissipation process and the concepts on di8erent stages of structural collision are not the unique features of the ring-on-beam collisions. Actually they characterize any structural collision, especially for the situation in which the structures do not exhibit hardening e8ect in their large deformation. One can refer to Ref. [10] for more detailed discussions on these concepts.

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780 0.8

1771

Ring Beam Local contact

Energy partitioning

0.6

0.4

0.2

0 0

(a)

1 2 3 Non-dimensional response time


4

Plastically dissipated energy portion

0.7

Ring Beam

0.6 0.5 0.4 0.3 0.2 0.1 0 0

(b)

0.5

1

1.5

2

2.5

Response time

Fig. 19. Energy dissipation history for the case in which: (a) hb =hr = 3, g = 10, and the beam is free in axial direction, (b) hb =hr = 1:2, g = 10, and the beam is axially constrained.

If the colliding structures exhibit deformation-hardening during large deformation, the kinetic energy remained at the restoring instant, K R , may not be dissipated only by one of the structures, since the instantaneously less sti8 structure (at the restoring instant) may increase its strength during further deformation and become sti8er than the other one. Thus, the dissipation of the remaining kinetic energy in both colliding structures could alternatively increase. Moreover, due to the inertia e8ect, the previously less sti8 structure will undergo over-hardening since the velocity does not vanish at the instant when the strength of the structure reaches the same level as the other structure (which is regarded as the sti8er one before this instant). When a structure is over-hardened, most of energy dissipation has to go to the other one until the latter is suBciently hardened. Thus, we may observe several steps in an energy dissipation process. Fig. 19(b) demonstrates a typical energy dissipation process for the case in which the beam is axially constrained, hb =hr = 1:2 and g = 10.

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Several steps in the energy dissipation process can be clearly observed, indicating that the beam or the ring is over-hardened so that the two structures alternatively dissipate energy at that period. 4.4. Mass and velocity e8ects on energy partitioning As analyzed above, if the less sti8 structure does not exhibit notable deformation-hardening, the energy partitioning in the two structures as well as the local contact deformation can be expressed as dlocal = Dlocal =K0 ;

(26a)

dsti8er = DR =K0 ;

(26b)

R dsofter = Dsofter =K0 + K R =K0 = 1 − dlocal − dsti8er ;

(26c)

where d denotes the portion of the initial kinetic energy dissipated by a structure (either sti8er one or softer one) or the local contact deformation, K0 is the initial kinetic energy carried by the ring, R Dsofter is the plastic energy dissipated by the softer structure until the restoring instant and Dlocal represents the plastic energy dissipated in the local contact deformation, DR and K R have been de7ned previously. It should be noted here that the summation of dlocal , dsti8er and dsofter is assumed to be equal to 1 since the initial kinetic energy is assumed much larger than the elastic capacity of the structures in the current analysis. The restoring energy measures the ability of a structure transforming from an energy-dissipation state to a non-dissipation state. Evidently if a structure’s elastic capacity is higher, the restoring energy will be smaller for a given energy dissipation state. An energy dissipation state herein pertains to a non-uniform velocity distribution caused by the local contact in the collision stage. Note that the severity of the local velocity change, i.e., the ratio of the velocity change to the initial velocity, depends on structural parameters (e.g., the mass and sti8ness in the local contact region) instead of the magnitude of the initial velocity. Hence the velocity distribution pattern at the end of the collision stage is independent of the magnitude of the initial velocity. Thus, a di8erent initial velocity will a8ect the magnitude of the restoring energy but may not notably alter the ratio of the restoring energy to the initial kinetic energy, dsti8er . As a result, it can be seen from Eq. (26) that the energy partitioning in the two structures will be independent of the initial velocity. Nevertheless the attached mass on the ring imposes no e8ect on the structure’s elastic capacity. By increasing the attached mass, the initial kinetic energy K 0 increases, resulting in a reduction of the ratio of the restoring energy to the initial kinetic energy (dsti8er ). Hence, it is seen from Eq. (26) that the increase of the attached mass on the ring will reduce the portion of the initial kinetic energy dissipated by the sti8er structure while it will increase the portion dissipated by the less sti8 structure. As an extreme case, if the attached mass is suBciently large, the energy partitioning will approach the static estimation, that is, all the kinetic energy will be dissipated by the less sti8 structure alone. Fig. 20(a) illustrates the mass and velocity e8ects on the energy partitioning for the case of hb =hr = 3 and the beam without axial constraint. In this case, the ring is a sti8er one as indicated in Fig. 18. The initial kinetic energy increases with the increase of the attached mass and the initial velocity, respectively. It is observed that with the increase of the initial velocity, the energy partitioning does not change notably. On the other hand, the increase of the attached mass results

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780 Beam (mass effect) Ring (mass effect) Beam (velocity effect) Ring (velocity effect)

1 0.9 0.8

Energy partitioning

1773

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

3

(a)

5

7

9

Non-dimensional input energy

1

Beam Ring Local Contact Total Plastic dissipation

0.9

Energy partitioning

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

(b)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Non-dimensional initial velocity v0

Fig. 20. The e8ect of the attached mass and the initial velocity on the energy partitioning for the case of hb =hr = 3; the beam is without axial constraint: (a) the initial kinetic energy varies with the attached mass or the initial velocity, (b) the initial kinetic energy remains constant while the attached mass and the initial velocity are varied simultaneously.

in a larger portion of the initial kinetic energy being dissipated by the beam. Fig. 20(b) shows that for the same case, if the initial kinetic energy is 7xed, with the increase of the attached mass (i.e., the decrease of the initial velocity), the energy partitioning pattern approaches the static estimation; that is, the beam dissipates all the initial kinetic energy. The independence/dependence of the energy partitioning on the initial velocity/attached mass has also been studied by Ruan and Yu [10] in analyzing the collision of mass–spring systems; and this is believed to be a common phenomenon for the collision between two structures if the structures do not exhibit notable deformation-hardening in their dynamic responses. The deformation-hardening could alter the relative strength of the two structures so that the remaining kinetic energy would not be simply dissipated by a single structure. Consequently the energy partitioning would not be as simple as given by Eq. (26). Fig. 21 shows the mass and velocity e8ects on the energy partitioning for the case in which the thickness ratio hb =hr is 2.5 and the beam is axially constrained. Fig. 22 is the corresponding quasi-static load–deformation behavior of the two structures. It can be seen that during large deformation, the beam is relatively sti8er due to the e8ect of axial tension. At the same time, the ring also exhibits deformation-hardening when its compression y does not exceed 0.32, although its load-carrying capacity decreases afterwards. Therefore, when the initial kinetic energy of the ring is small, the hardening e8ect of the ring makes the beam

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H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780 1

Beam (mass effect) Ring (mass effect) Beam (velocity effect) Ring (velocity effect)

0.9

Energy partitioning

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

Non-dimensional initial kinetic energy

Fig. 21. The e8ect of the attached mass and the initial velocity on the energy partitioning for the case of hb =hr = 2:5; the beam is axially constrained. Simply supported beam with axial constrain

p 40

Ring

30

20

10

0 0.5

0.4

0.3

0.2

yb

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yr

Fig. 22. The quasi-static load–deformation behavior of the ring and the axial constrained beam (hb =hr = 2:5).

dissipate a portion of the remaining kinetic energy. Moreover, as the initial kinetic energy increases either by increasing mass or velocity, the portion dissipated by the beam also increases. However, when the initial kinetic energy is suBciently large (e.g., the non-dimensional value of the initial kinetic energy is larger than 3), the energy partitioning varies very little if the increase of the initial kinetic energy is due to the initial velocity. This is because in this case the beam is suBciently hardened after it dissipates the resorting energy and thus all the remaining energy will go to the ring. Hence the initial velocity does not notably alter the energy-partitioning pattern. On the other hand, as the initial kinetic energy increases due to the attached mass on the ring, in a larger energy level, the less sti8 structure (the ring) will dissipate a larger portion of the initial kinetic energy, as illustrated in Fig. 21. If the thickness ratio is small (e.g., hb =hr = 1), it is found that the 7nal con7guration of the ring is in a “U” shape. Actually, in this shape the ring is greatly hardened, since the vertical “columns” of a “U” shape is very sti8 in its axial compression. Consequently, not only all the remaining kinetic energy will be dissipated by the beam, but also the restoring energy of the ring is very small. As shown in Fig. 23, with the increase of the initial kinetic energy, the portion of the ring’s dissipation continuously decreases, except in the case of low initial kinetic energy (e.g., the non-dimensional

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1775

1 0.9

Energy partitioning

0.8 0.7

Beam (mass effect) Ring (mass effect) Beam (velocity effect) Ring (velocity effect)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15

20

25

Non-dimensional initial kinetic energy

Fig. 23. The e8ect of the attached mass and the initial velocity on the energy partitioning for the case of hb =hr = 1; the beam is axially constrained.

20

The non-dimensional Energy

18

The non-dimensional energy dissipated by the ring

16 14 12 10 8 6 4 2 0 0

5

10

15

2 The non-dimensional initial kinetic energy ( g + π )v0 2

20

Fig. 24. The non-dimensional energy dissipation of the ring, varying with the initial kinetic energy for the case of hb =hr =1; the beam is axially constrained.

initial kinetic energy is smaller than 4) the portion of the ring’s dissipation increases since the ring is not suBciently hardened. Fig. 24 shows that for a U-shaped deformed ring, the corresponding energy dissipation exhibits little increase as the initial kinetic energy greatly increases, especially at the high-energy level. 5. Conclusions With the collision between a vehicle and a roadside parapet in mind, the normal collision between a ring and a beam is studied. The beam is assumed to be simply supported with or without axial constraint. A rigid mass block is attached to the half ring to model the undeformable parts of a vehicle. A mass–spring 7nite di8erence model (MS–FD) is employed and enhanced to incorporate large deformation and axial stretching (or compression). In general, the results obtained from the model

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H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

show that the less sti8 structure dissipates more energy. However, the dynamic energy partitioning cannot be simply estimated from the quasi-static behavior of the structures. The very localized velocity redistribution in the contact region is the key feature of the dynamic collision process, which makes the dynamic energy partitioning quite di8erent from the quasi-static one. Accordingly, the energy can be separated into a dynamic part and a quasi-static part. The attached mass more signi7cantly a8ects the energy partitioning between the ring and the beam, since the quasi-static part increases as the mass increases. Axially constrained beam is structurally hardened during deformation, thus the energy-partitioning pattern is more complicated. In general, at higher initial kinetic energy level, the beam is greatly hardened and becomes much sti8er than the ring. However, if the beam is thin, the ring can deform into a “U” shape, leading to a signi7cant hardening of the ring. The local energy dissipation is very small compared with the initial kinetic energy, while its magnitude depends only on the initial velocity. However, the evolution of the contact region does a8ect the global deformation and energy-partitioning pattern. Moreover, the multi-collision phenomenon is observed from the analysis of the structural collision model. Acknowledgements The work described in this paper was conducted as a part of CERG research project HKUST 6035/99E funded by the Honk Kong Research Grant Council (RGC). This support is gratefully acknowledged. Appendix Refer to Ref. [9], the deformation process of an end constrained half ring quasi-statically compressed by a rigid plate as shown in Fig. 18 can be divided into three phases. Due to symmetry, only a quarter of the ring is considered. As shown in Fig. 25, Phase I is a three-stationary-hinge phase, in which the hinge 1 locates at the mid-point A of the half ring, the hinge 2 locates at the supporting point B, and the hinge 3 is at the mid-point C of arc ACB. For a given value of  I (shown in Fig. 25), The chord length . (A.1) |BC| = 2R sin ; 8 where  denotes the length of the segment. The angle /I (shown in Fig. 25(a)) is obtained from

 R − |BC| cos 3.=8 +  I |A C| I = arccos : (A.2) / = arccos |BC| |BC| The angle  I (shown in Fig. 25) is obtained from . 3 I  = − /I − .: 2 8

(A.3)

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

1777

P

A' A (1) φI

(2) C

3π/8 xr

B (3)

H

A''

θ' I

ϕI

H' θ I Yr

O'' O

Fig. 25. The quasi-static deformation mechanism of an end-constrained half ring during Phase I.

The angle ’I (shown in Fig. 25) is obtained from

  |BC| sin 3.=8 +  I + |BC|2 − |A C|2 |HH | + |AA | I = arctan ’ = arctan ; R − |A H| R − R sin  I where |A C| = R − |BC| cos(3.=8 +  I ). Thus, the displacement of the rigid plate Yr is given as



3 3    I I . − / − |BC| sin .+ : Yr = O H − O H = R sin 8 8 The distance xr of the line of resultant force from B or C is thus

3 |BC| I I sin .+ −’ : xr = 2 8 Hence, the applied load P is given as Mpr P=2 sin ’I : xr

(A.4)

(A.5)

(A.6)

(A.7)

Phase I ends when the hinges at A and C lie on the same horizontal line (i.e., A and A coincide each other), and the angle

3. R − |BC| I − : (A.8)  = arccos |BC| 8 During Phase II, the kinematic conditions require the hinge 2, at C (Fig. 26(a)), to remain at the same horizontal level as hinge 1 at A. Fig. 26(a) and Fig. 26(b) sketch the two successive steps in Phase II, and is indexed i and i + 1, respectively. Following Reddy and Reid [9], the travel of the hinge 2, at C, is discretized as follows: the hinge at C travels down to C through a distance 2R II and remains at the same horizontal level as point A (Fig. 26(b)). The arc CC of length 2R II is

1778

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780 P

P

(2) C

H

(π − 2ψ

A (1)

II i

)

C

(2) C'

I

− ψ II − θ iII − θ II 2 + φ II 2

φ iII

φ II

H

A (1) φiII − φ II

ϕ II

ψ iII (3) B

ψ iII + ψ II

O''

H'

xr B

O

θ iII

(3)

θ iII + θ II

O'

H'

O'' O

Yr

O'

(a)

(b)

P Yr

M N 3π 8 − φ0III

A(1)

(2) C

φ (φ ) II k

III 0

ψ (ψ ) II k

(3)

B

(c)

III 0

θ kII (θ 0III )

O'' O

O'

Fig. 26. The quasi-static deformation mechanism of an end-constrained half ring during Phase II: (a) step i during Phase II, (b) step i + 1 during Phase II, (c) the end of phase II (also the start of Phase III).

replaced by the in7nitesimal rigid chord CC having plastic bending moment Mpr at each end. The travel of the hinge 2 from C to C is a8ected by the equal rotations of (/II =2 +  II =2) at C and C , respectively, and a rotation of /II occurs at A. The line of resultant force is parallel to BH and is equidistant from B and C . Suppose all the geometrical parameters in step i (Fig. 26(a)) are known, the following geometrical relation in step i + 1 (Fig. 26(b)) can be formulated. Note that the segment CI is common to triangles CC I and CAI, it gives

|AC | sin /II = |CC | cos . − 2 iII − iII −  II =2 −

= R II cos . − 2 iII − iII −  II =2 − where

i

II

 + /II =2 ;  II + /II =2 ;

, iII and /IIi are de7ned in step i, as shown in Fig. 26(a).

(A.9)

H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

Since |AC | = |AI| + |IC|, we have

 2R II sin . − 2 iII − iII −  II =2 − II + /II =2 + |AC | cos /II 

=|OB| − |BC | cos iII + II + iII +  II



 =R − 2R cos iII + II cos iII + II + iII +  II :

1779

(A.10)

For a given in7nitesimal II , the angles  II and /II can be obtained by solving Eqs. (A.9) and (A.10). The displacement of the rigid plate Yr is then obtained from  



(A.11) Yr = O H − O H = R sin /IIi − /II − |BC | sin iII + II + iII +  II : Note that the point C and H are in the same horizontal line, the incline angle ’II of the line of resultant force is obtained from 

II 

II II II II II sin + + +  +  2R cos i i i 

: (A.12) ’II = arctan (R − R cos /IIi − /II ) The distance xr of the line of action of resultant force from B or C is  |BC | II sin i + II + iII +  II − ’II ; 2 



= R cos iII + II sin iII + + iII +  II − ’II :

xr =

(A.13)

Hence, the applied load P is given by Eq. (A.7). II II II II To solve the next step i+2, the angles i+1 , i+1 , and /IIi+1 is given by i+1 = iII + II , i+1 =iII + II  and /IIi+1 = /IIi + /II , respectively, and the chord length |AC| is updated by |AC |. The initial condition of Phase II is obtained from the end of Phase I, which give

3 3 R − 2R sin .=8 3 II II .  − .; /II0 = .: (A.14) = = arccor 0 0 8 2R sin .=8 8 8 Phase II ends at kth step when the angle /IIk = .=4, that is, the rigid plate in contact with the mid-point of the quarter of the ring, M, as shown in Fig. 26(c). Since the line NM tangent to the curve BCM at M is not horizontal (Fig. 26(c)), the contact point will remain at the point M in Phase III. This is also observed from the MS–FD solution. The hinge 2, at C, continues to travel down towards the supporting point B. Phase III thus ends when the hinge 2 reaches the point B. The discretization of the traveling hinge is shown in Fig. 27, and the solution in Phase III is the same as that in Phase II, except that in each step (say step i), the incline angle of the line of resultant force ’III , and the deformation of the ring Y are respectively given as (Refer to Fig. 27)



 2R cos iIII sin iIII + iIII III ;

(A.15) ’ = arctan R − 2R sin(.=8) cos 3.=8 − /III i 





− 2R cos iIII sin iIII + iIII : (A.16) Yr = R − 2R sin(.=8) sin 3.=8 − /III i

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H.H. Ruan, T.X. Yu / International Journal of Mechanical Sciences 45 (2003) 1751 – 1780

A'

P Yr M

3π 8 − φiIII

C (2) C'

A (1)

ϕ III

ψ iIII (3) B

θ iIII

φ iIII O'' O

O' Fig. 27. The quasi-static deformation mechanism of an end-constrained half ring during Phase III.

References [1] Yu TX, Yang JL, Reid SR. Deformable body impact: dynamic plastic behavior of a moving free-free beam striking the tip of a cantilever beam. International Journal of Solids and Structures 2001;38:261–87. [2] Ruan HH, Yu TX. Local deformation models in analyzing beam-on-beam collision. International Journal of Mechanical Science 2003;45(3):397–423. [3] Lee EH, Symons PS. Large plastic deformations of beams under transverse impact. Journal of Applied Mechanics 1952;19:308–14. [4] Parkes EW. The permanent deformation of a cantilever struck transversely at its tip. Proceedings of the Royal Society of London, Series A 1955;228:462–76. [5] Stronge WJ, Yu TX. Dynamic models for structural plasticity. London: Springer; 1993. [6] Jones N. Structural impact. Cambridge: Cambridge University Press; 1989. [7] Hou WJ, Yu TX, Su XY. Elastic e8ect in dynamic response of plastic cantilever beam to impact. Acta Mechanica Solida Sinica 1995;16:13–21. [8] Huang X, Yu TX, Lu G, Lippmann H. Large de0ection of elastoplastic beams with prescribed moving and rotating ends. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science 2003;217:1001–4. [9] Reddy TY, Reid SR. Lateral compression of tubes and tube-systems with side constraint. International Journal of Mechanical Science 1979;21:187–99. [10] Ruan HH, Yu TX. Collision between mass–spring systems. International Journal of Impact Engineering, in press.