22陈OntheImpactforceandEnergyTransationduringShip.doc

On the Impact force and Energy Transformation during Ship-Bridge CollisionsLili Wang1,* Corresponding author. E-mail address: (Lili. Wang) , Liming Yang1, Changgang Tang1, Zhongwei Zhang1, Guoyu Chen2 and Zonglin LU31, Mechanics and Material Science Research Center, Ningbo University, Ningbo, 315211 China;2, Shanghai Marine Steel and Structure Research Institute, Shanghai, 201204, China; 3, Department of Bridge Engineering, Tongji University, Shanghai, 200092, ChinaABSTRACT: Taking ship-bridge collision as an example, how the wave propagation and the dynamic behavior of materials influence the impact force and energy transformation are studied from the impact dynamics view-point. By using a new flexible, energy-dissipating crashworthy device developed by the authors, both the impact force and energy transformed can be markedly reduced, which is much conductive to protect both bridge and ship. Keywords: Impact force; energy transformation; ship-bridge collision; wave propagation; high strain rate1. INTRODUCTIONIn the studies of dynamic response of structures and materials under impact loading, two dynamic effects, i.e. the inertia effect and the strain rate effect, should be taken into consideration. The study of the former results in analyses of wave propagation in various forms, either explicitly or implicitly; and the study of the latter has promoted the research of all kinds of nonlinear rate-dependent constitutive relations and failure criteria.Correspondingly, those two dynamic effects should be taken into account in the analyses of impact force and dynamic energy absorption/transformation of structures under impact loading, such as those take place during ship-bridge collisions.In fact, considering a typical structure with a characterized scale of Ls under a transiently applied loading with a characterized duration TL, if the velocity of wave propagating within the structure is characterized by Cw, then the dynamic response time of the structure can be characterized by TW (=Ls/Cw). Note that the characterized wave velocity Cw is strongly dependent on the dynamic physical-mechanical properties of materials. Obviously, when the dimensionless time T*(= TL/TW) 1 (called the late response of structures), and then the analysis of the detailed process of wave propagation is no longer needed. In the present paper, the attention is mainly focused on the early response of structures. Taking ship-bridge collision as a typical example, how the wave effect and the dynamic behavior of materials influence the impact force and the energy absorption/transformation will be studied.Finally, a typical engineering example of ship-bridge collision is analyzed and discussed.2. IMPACT FORCEAccidents due to ship-bridge collision have taken place often since bridges appeared, and become more serious with increasing huge ship and large bridge 1-4. More attention should be paid to the techniques to avoid the increasingly serious accidents resulted from ship-bridge collisions.For both ship-designers and bridge-designers, the key should be known is how the impact force in the process of ship-bridge collision can be determined scientifically.In China, like in other countries, two typical formulas are currently recommended to estimate the impact force during ship-bridge collision: one is based on the momentum principle (or impulse principle), which is usually expressed as 5;(1)and another is based on the kinetic energy principle, which is usually expressed as 6,(2)where F is the impact compressive force (MN), W and m the weight (MN) and the mass of ship respectively, v the ship impact velocity (m/s), t the impact duration, a the impact angle between the ship navigation direction and the bridge pier surface, C1 and C2 the elastic flexibility (m/MN) of the ship and the bridge pier respectively, g (s/m1/2) the reduced coefficient of kinetic energy, considering that the kinetic energy of ship is not totally absorbed by bridge.When eqn (1) is adopted, the most difficult is how to correctly determine the impact duration t in different process of ship-bridge collision. In practice, designers have to use some empirical data, which are lack of theoretical or experimental evidence and consequently may lead to large errors. Thus, more designers prefer to adopt eqn (2).Theoretically, eqn (1) and eqn (2) are inherently interrelated. If the relative displacement between the ship and bridge pier during collision is denoted by U, then the average t in eqn (1) can be calculated from U divided by the impact velocity v, namely, t=U/v, and consequently eqn (1) can be re-written as . (1a)The physical meaning of eqn (1a) is that the work done by the impact force is equal to the kinetic energy of ship, another expression of kinetic energy principle. Furthermore, for an elastic system, the displacement U resulted from an impact force F is proportional to the impact force, U=CF, where C is the elastic flexibility (the reciprocal of the rigidity K) of elastic system, thus eqn (1) can be further re-written as . (1b)Taking into account the influence of impact angle a in the case of oblique collision, considering only a b (=gg1/2) part of kinetic energy of ship being absorbed by the bridge, and taking C as the sum of the flexibility C1 of ship and the flexibility C2 of bridge, C=C1+C2, eqn (1b) exactly coincides with eqn (2).However, as can be seen in the above analysis, both eqn (1) and eqn (2), as well as other similar equations recommended in most existing codes or guide specifications 7, 8, are all based on simplified, quasi-static, elastic analyses, taking ship and bridge respectively as a whole body, without any dynamic consideration of wave propagation and rate-dependent plastic deformation for both ship and bridge.Although the speed of a ship when it passes through a bridge is generally requested to be decreased to several meters per second, it should be emphasized that collision between a ship and a bridge is a complicated dynamic process. In fact, the ship dimension is now as big as in the order of 102 m, the ship mass is as large as in the order of 106-108 kg, and thus the kinetic energy of a ship is as large as in the order of 102 MJ, so the ship-bridge collision is a dynamic process with huge energy exchange taking place in a short duration (in terms of 102 millisecond to few second), particularly when the early response of ship and bridge are concerned. Therefore it should be solved by an impact dynamics approach, not by a quasi-static approach as traditionally used.From a view based on wave propagation theory 9, 10, the dynamic impact force at the collision interface is complexly determined by the coupled process of wave propagation within both of ship and bridge under a given initial-boundary conditions. However, the specific process of wave propagation within ship/bridge is further dependent on their structural details and the constitutive behavior of the material used. In fact, in the case of one-dimensional (1D) elastic-plastic stress wave propagation, it is well-known that the governing equations are constituted by the following equations 10, i.e. the motion equation (momentum conservation equation),(3)the continuity equation (mass conservation equation),(4)and the constitutive equation of material at high strain rate,(5)where r0 denotes the density, s the stress (defined as positive for tensile case), e the strain (defined as positive for tensile case), and v the particle velocity (defined as positive when the particle moves in the x axis). By means of the well-known characteristics method 10, the above partial differential equations are equivalent to two sets of ordinary differential equations, each set consisting of a characteristics equation and a corresponding compatibility condition along the characteristics:,(6) or ,(7)where the positive sign is for rightward waves, the negative sign for leftward waves, Cw the wave velocity along the characteristics, ,(8)and r0Cw is called “wave impedance”.For rightward simple waves, satisfying the boundary condition of constant v impact, the solution can be obtained from eqns (6) and (7) as 10:,if s sy; (9a), if s sy, (9b)where sy is the yield stress, Cwe the elastic wave velocity, Cwe= (E/r0)1/2 which is a constant, and E the Youngs modulus. Noticing F=-sA, where A is the area of collision interface, eqn (9b) can be re-written as, in elastic cases. (9c)Therefore, even in the elastic cases, as can be seen by comparing the dynamic eqn (9c) and the quasi-static eqn (1b), although the impact force F is identically proportional to the impact velocity v in both equations, an essential difference exists between those two equations. In the quasi-static analysis (eqn 1b), F is proportional to the square root of the total mass m and structural rigidity K (=1/C) of ship, while in the dynamic analysis (eqn 9c), F is proportional to the generalized wave impedance r0CweA (or the square root of the density r0 and elastic modulus K of material). It means when the early response of ship-bridge collision is concerned, the material-dependent wave impedance plays a dominant role, rather than the total mass and rigidity of ship. It is not difficult to understand, only with the wave propagating forth and back repeatedly within the ship and the bridge, will the total mass of ship gradually influence the impact loading and the energy exchange.If a collision process between two elastic bodies, such as ship-bridge collision, is concerned, it is a coupled and interacted process of waves propagating within both of ship and bridge. Thus the impact stress ss-b will depend on both the ship wave impedance (r0Cw)s and the bridge wave impedance (r0Cw)b. Satisfying the conditions of stress balance and displacement continuity on the collision interface, the following solution can be obtained from eqns (6) and (7) 10:. (10)Introducing a dimensionless impact stress defined as = -s /(r0Cw)sv), eqn (9b) can be simplified to =1 while eqn (10) can be simplified to =1 /(1+ns-b), where ns-b is the wave impedance ratio, ns-b = (r0Cw)s /(r0Cw)b. Comparing those two equations, it is clear that the wave impedance ratio ns-b now plays a dominant role. Obviously, with increasing ns-b, the impact stress ss-b decreases. On the contrary, when (r0C)b approaches infinite, the ns-b approaches zero, eqn (10) is reduced to eqn (9b), which corresponds to the extreme situation, namely the impact stress reaches its maximum when the ship impacts onto a rigid wall.To quantitatively illustrate how the impact stress depends on different wave impedance ratio, we consider a simplified bar-bar collision system (Fig. 1), where the finite bar Bs (simulating the ship with a length of L) is made of steel (with r0=7.85x103 kg/m3, E=210 GPa, n=0.3 and thus Cw=5.17 km/s and r0Cw=40.6 MPa s/m), while three different materials are considered for the bar Bb (simulating the bridge pile but neglecting the reflected waves from another end). They are (a) rigid material (with r0Cw=), (b) the same steel as for Bs and (c) concrete (with r0=2.50x103 kg/m3, E=25 GPa, n=0.17 and thus Cw=3.16 km/s and r0Cw=7.9 MPa s/m). When the bar Bs axially impacts onto the bar Bb with velocity 5 m/s, the impact stresses on the collision interface for those three different cases of pile material are calculated by eqn (10) and shown in Figure 1(a), (b) and (c) respectively. The numerical results simulated by the LS-DYNA code are given in the same figures, showing coincide results. The comparison of those three cases is given in Figure 1(d). As can be seen, the ratio of impact stress among those three cases is as large as 2 : 1 : 0.32, since the ratio of steel wave impedance and the concrete wave impedance is as large as 5.14. It means that the higher the wave impedance of bridge pile is, the larger the impact stress will be. In other words, any rigid protection for bridge pile will induce larger impact loading, which is not conducive to the bridge safety, but also not conducive to the ship safety.(a)(b)(c)(d)Figure 1. Impact stress calculated for the collision between (a) steel body to rigid body, (b) steel body to steel body, (c) steel body to concrete body, and (d) the comparison between those.Thus, it is clear that if a flexible protection device with lower wave impedance, rather than with higher stiffness, is built surround the bridge pier, it can decrease the impact force and should be conducive to the safety of both the bridge and the ship. Moreover, if such a protection device is characterized not only by flexible (low wave impedance) response but also by energy-dissipating response, it should be more conducive to the safety of both the bridge and the ship. To quantitatively illustrate how the protection device with flexible and energy-dissipating behavior influences the impact stress, we further consider a simplified “bar Bs-damping layer Bd bar Bb” collision system (Fig. 2), where a flexible damping layer Bd connects with the bar Bb and is made of polymer with r0=1.19x103 kg/m3. The dynamic behavior of this polymer damping layer is modeled by a three-element visco-elastic body with E=2.94 GPa for the paralleling elastic element, elastic modulus EM=3.07 GPa and relaxation time qM=95.4 ms for the Maxwell element. Its characteristic wave velocity is Cw=(E+EM)/r0)1/2=2.25 km/s, and correspondingly the transient wave impedance is r0Cw=2.68 MPa s/m and the wave impedance ratio of steel to polymer is as large as 15.1.With such a damping layer Bd (modeling a flexible and energy-dissipating protective device) the impact stresses for the previously mentioned three different cases of pile material are calculated by LS-DYNA code. The numerical results are given in Figure 2(a) and the comparisons between those results with the results shown in Fig.1 are given in Figure 2 (b). Obviously, the existence of damping layer greatly decreases the impact stress, regardless what material is. On the other hand, the impact duration is prolonged, accompanied with viscous energy dissipation.(a)(b)Figure 2. Impact stresses calculated (a) for three cases mentioned in Fig. 1 but with polymer damping layer, (b) comparisons of impact stresses calculated with and without damping layer.As can be imaged from the above analysis, distinguished from the traditional design of constructing a protective device with high stiffness and strength, the use of a flexible and energy-dissipating protection device will definitely and greatly decrease the impact force, and simultaneously prolong the collision duration. Note that the prolongation of collision duration under low impact force will enable the ship to have enough time to turn its navigation direction and consequently as much of remaining kinetic-energy can be carried off by the turned-away ship. So, it will be further conducive to the safety of both the bridge and the ship, which will be discussed in detail in the later section. 3. ENERGY ABSORPTION AND TRANSFORMATIONThe process of ship-bridge collision is a dynamic process of energy transformation between ship and bridge in a short duration. However, in the quasi-static deduction of eqns (1) and (2), it was implicitly assumed that the overall mass of the ship as a whole participates the momentum and energy exchange, fully neglecting the time-dependent process of wave propagation. In fact, according to the wave propagation theory 10, neither the whole ship nor the whole bridge are immediately involved in the energy transformation, but only those parts of mass behind the propagating wave front are involved in the momentum and energy exchange. In other words, the range of the mass involved in the energy transformation is developing with wave propagating. Obviously, the faster the wave velocity is, the more extensive the range of energy transformation is. On the other hand, the wave velocity is closely related to the dynamic behavior of material (cf. eqn 8). Thus, from the view-point of impact dynamics, the energy absorption/transformation is closely dependent on both the wave effect and the material strain rate effect. To illustrate those effects, the early responses of a long bar suffered impact loading, made of different materials (including elastic, elasto-plastic. and viscoelastic), are analyzed. Consider a wave with strong discontinuity front, propagating rightward with wave velocity D. Across such a wave front, the well-known Rankine-Hugoniot relationships hold, which consist of three conservation conditions 10, namely the momentum conservation condition , (11)the mass conservation condition, (12)and the energy conservation condition, (13)where e denotes the materials internal energy in a unit mass (or r0e is the internal energy in a unit volume) and the notation denotes the jump of a quantity across the wave front, e.g. v=v-v+, where v- and v+ denote the value of v just after and before the wave front, respectively. Substitute eqns (11) - (12) to eqn (13), after some mathematical manipulations we obtain. (13a)In other words, the jump of internal energy across the wav