Contents lists available at ScienceDirect Marine Structures journal homepage: www.elsevier.com/locate/marstruc On multi-planar impact mechanics in ship collisions Zhenhui Liu a,∗ , Jørgen Amdahl b a Front End, Aker Solutions AS, NO-7044, Trondheim, Norway b Centre for Autonomous Marine Operations and Systems, Department of Marine Technology Norwegian University of Science and Technology, NO- 7491, Trondheim, Norway ABSTRACT This paper presents a new solution for ship impact dynamics in multi-planar space. A collision matrix is presented. It is derived on the basis of impact dynamics for rigid bodies. Effective mass factors are introduced accordingly. An impulse-based solution for ship collisions in multi-planar space is obtained. The hydrodynamic effects are included by means of constant added mass factors. Friction effects are considered; thus, “stick” and “slide” mechanics are both captured. The present method is compared with other models for ship collisions. It is shown that the method unifies existing methods for the normal direction [1], planar space [2] and multi-planar space [3]. Application examples are shown to demonstrate the benefits of the method. Discussions and clarifications of some important concepts of the theory are included. 1. Introduction Ship collision is an important topic for the maritime and offshore industries due to its severe consequences. The “Sanchi” oil tanker accident in the East China Sea is a recent example. The study of impact dynamics is fundamental for understanding the complicated collision process. It can be used for consequence assessment, risk assessment and ice load calculations, as discussed by Wang et al. [4], Pedersen [5] and Dolny [6]. The energy released by rupture and plastic deformation in a collision is of great interest. Petersen [7] presented a numerical model of ship collision dynamics in the time domain. Strip theory was used to take the hydrodynamic forces into account. Pedersen and Zhang [2] developed a closed-form solution for ship collisions with motion restricted to the plane of the water surface (a 3DOF +3DOF system). The effect of friction was considered in their work. The dissipated energy was obtained, and it was shown that the dissipated energy varies with the attack angle and the impact location. Constant added mass factors were simply utilized to include the influence of hydrodynamic forces. Caution should be paid when building the global and local impact coordinate system; otherwise, the results obtained may be totally different (see discussions in this paper). Liu and Amdahl [3] presented a new for- mulation to describe the ship impact mechanics in 3D space. It originates from the impact theory of Stronge [8]. The new metho- dology solves the problem in a full 6DOF+6DOF system. It downgrades to a 3DOF+3DOF system if the vertical eccentricity is ignored [3]. The key idea of this method is to formulate the equation of motion in a local coordinate system located at the impact point by utilizing the inverse inertia matrix provided by Stronge [8]. The hydrodynamics coefficients have to be transformed into this local coordinate system. The “equivalent mass” concept was adopted on a general basis by Liu and Amdahl [3]. Yu et al. [9-10] compared the method of Liu and Amdahl [3] with advanced numerical simulations. They concluded that the analytical method yields a good prediction of the total strain energy dissipation during the first impact period. Zhang et al. [11] re-examined the validity of the 3DOF+3DOF method [2]. They compared model scale experiments reported by Tabri et al. [12]-[13] with analytical results. It was concluded that the method agrees well with the tests. A model that includes roll motion was presented at the same time. Popov et al. [1] presented a 6DOF+3DOF model for ship and ice floe collisions. It was assumed that the impact impulse always https://doi.org/10.1016/j.marstruc.2018.10.006 Received 7 March 2018; Received in revised form 2 October 2018; Accepted 9 October 2018 ∗ Corresponding author. E-mail address: zhenhui.liu@akersolutions.com (Z. Liu). Marine Structures 63 (2019) 364–383 0951-8339/ © 2018 Elsevier Ltd. All rights reserved. T