PSI - Issue 82

Ivan Ćatipović et al. / Procedia Structural Integrity 82 (2026) 302 – 308 I. Ćatipović et al./ Structural Integrity Procedia 00 (2026) 000–000

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2. Mathematical model 2.1. Motion equation

The motion equation for analysing mooring lines is documented in Ćatipović et al. (2021a, 2021b, 2012, 2011). It uses a spatial curve to show the deformed shape of a mooring line, with a time-dependent position vector r defining the curve in a three-dimensional coordinate system. An arc length s of a non-extended mooring line represents a point on this curve. The equation is expressed as

! ! ! " ! ! # & " # & + (

$ ' )

& !

A % ! % " " # ! !! A + =

(1)

+

"

where T E is the effective tension force (API RP 16Q, 2017; Sparks, 2018). The first equation term represents the geometric stiffness forces of a mooring line. The axial deformation is denoted as e , and the acceleration as . The right side gives the inertial force based on the distributed mass m . Vector g is the gravitational acceleration, so m g is the line's weight. Seawater density ρ is used to evaluate the apparent buoyancy, ρA g , according to Sparks (2018), where A is the line's cross-sectional area. The term q H accounts for hydrodynamic loads, formulated via Morison equation in Ćatipović et al. (2021b). 2.2. Stiffness of synthetic mooring ropes Figure 1 shows the typical tension-extension characteristics for conventional polyamide ropes from OCIMF (2008). It depicts the tension-to-minimum braking load (MBL) ratio versus extension e . ! !!

! +,-.

!"! !"# !"$ !"% !"& !"' !"( !") !"*

!

!"!!

!"!'

!"#!

!"#'

!"$!

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Fig. 1. The typical broken-in tension-extension characteristics for conventional polyamide ropes obtained from OCIMF (2008).

Following OCIMF procedures, the broken-in characteristics result from 10 cycles to 50% of rated strength, simulating long-term elasticity changes under low tension (OCIMF, 2008; 2018). These are measured on the tenth cycle, and it is assumed that the curves apply to loading rates over 1 minute. 2.3. Finite element formulation The main goal is to enhance the FE model's handling of stiffness forces, particularly through the geometric stiffness matrix. The typical data for the specific rope type (from OCIMF) is used to model the non-linear behaviour of the rope. Here, the data are represented by a function f that is used to calculate the axial deformation, i.e. extension, based on the real tension T acting on the rope as