Page 310 - Marine Structural Design

286 Part II Ultimate Strength

14.2 Finite Element Formulation
14.2.1 Equations of Motion
The striking and the affected structure are considered one structural system connected by
spring elements.
The equations of motion for the structural system are established under the following
assumptions:
The striking ship is treated as a rigid body without volume, and all the deformations in the
ship are assumed to take place in a zone around an impact point.
The deformations in the ship and the local indentation in the affected member of the
offshore structure are simulated by using nonlinear spring elements in which only
compression forces act. The force deformation curves for those spring elements are
functions of the strain-rate.
The deformation of the affected structure, except for the local indentation in the hit
member, is taken into account by using a model of the structure, which is composed of
three-dimensional beam-column elements.
The hydrodynamic forces acting on the ship are accounted for, by introducing an added
mass concept. Morison's Equation is applied with the purpose of including the fluid-
structure interaction to the affected structure's analysis.
When considering the dynamic equilibrium of the structural system, the equations of motion
may be written in an incremental form as:

[Ml{d4+[Cl{4+ [fGlb>= {@dl (14.1)
where {du}, {dzi} , and (dii} are the increments of nodal displacements, velocities, and
accelerations, respectively. [MI is a structural mass matrix, [q is a structural damping matrix,
and [K,] denotes the structural tangent stiffness matrix. The external load vector (dF, ] is due
to the drag force term in Morison's equation, which is evaluated using an approach described
by Bai and Pederson (1991). The added mass term in Morison's equation is included in the
structural mass matrix [MI.
The equation of motion, Eq. (14.1) is solved by using the Newmark-p method.
14.2.2 Load-Displacement Relationship of the Hit Member

In this section, a derivation of a nonlinear spring element will be used to model the local
indentation in the hiffaffected member. The spring element will be used for steel platforms and
the hiffaffected members are therefore assumed circular thin-walled tubes.
The linear elastic displacement of the load point for a pinch loaded tubular member can be
determined by:

(14.2)

where P is the force and 6, denotes the elastic displacement, E is Young's modulus, and T is
the thickness of the tube wall. D denotes the outer diameter of the tube while L, is the
characteristic length of the contact area along the axial direction of the tube.