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

306

5

where ¢ denotes spatial derivative along a line length. It is important to highlight that in the updated form of the motion equation, Eq. (9), two nonlinearities are managed. The first nonlinearity stems from the non-linear tension extension relationship of a synthetic rope. The second source of nonlinearity is the stiffness formulation itself. The term 1/(1+ e ) is nonlinear. The discretisation is achieved by the Galerkin method as presented in Ćatipović et al. (2021b, 2011). The is the stiffness matrix is defined from the first term of Eq. (9) as

! !"#$% &

!

!" = + !"#$%

!#

!!

!$

(10)

!"#$% ) )

)

)

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

!

+

! !

+

& !&"#$%

& ' !&'"#$%

with

" $ !

" $ !

# !

!"

!#

B & + , , - * # # " A% & '

B & ++ , , - " # # " # A% & '

*

=

=

! "

"

"#A%&'

"#A%&'

'()

(11)

" $ !

" $ !

$ !

!

!!

!$

!

B & ++ + , , - * " # # " # ( A% & '

B ++ + + , , - " # # " # ( ) A% & '

*

=

=

"#(A%&'

"#()A%&'

!

$

(

)

(

)

'()

'()

where is the Kronecker delta, and L is the element length. A k represents shape functions for modelling the displacements, while P n models the effective tension force along the element. Here, it is essential to note that the integrals in Eqs. (11) can be resolved analytically, enabling the analytical formulation of the stiffness matrix. 3. Case study This case study uses a single rope to demonstrate the efficiency of the enhanced finite element formulation. The rope measures 555 m in length with an MLB of 885 t. It is modelled with a single finite element. Two rope types are considered: Polyamide 3&8 Strand and Polyamide Double Braided. Their tension-extension characteristics are defined according to OCIMF (2008), as shown in Figure 1. The tension values are calculated using Eq. (10) and compared to the original data from OCIMF. For Polyamide 3&8 Strand, an approximation using 4 polynomials follows the Newton interpolation scheme, as outlined by Eqs. (5) to (7). The comparison results are shown in Figure 2, along with the relative errors in the calculated tension values relative to the original values. The same rope type is modelled using 8 polynomials for approximation, as shown in Figure 3. Finally, the Polyamide Double Braided rope is examined using 8 polynomials as presented in Figure 4. Table 1 summarises the relative errors for all three approaches concerning tensions above 5% of MBL. This lower limit for tension is based on DNVGL-OS-E301 (2015), which requires that the pretension in mooring lines be at least 10% of the MBL to prevent snapping loads. !" !

! ,-./

A11C1M7E

#F# #F$ #F" #F' #F( #F% #F) #F* #F+

$# $% "#

# %

!"# !$% !$# !%

#F##

#F#"

#F#(

#F#)

#F#+

#F$#

#F$"

#F$(

#F$) !

0123245678569:;

0<=524:L7

B:65=28:7:11C1

Fig. 2. Comparison of the original and calculated tension-extension relation using 4 polynomials for Polyamide 3&8 Strand rope.