PSI - Issue 82

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

305

4

!"# !

" &

# % '

(2)

" !

= $

In the motion equation, Eq. (1), the effective tension T E defines the geometric stiffness, while Eq. (2) relates to the real tension T . The connection between the effective and real tension in a mooring line is described in Sparks (2018) and API RP 16Q (2017). A simplification is used where the effective and real tension are considered equal, i.e., T = TE , as justified in Ćatipović et al. (2011). Thus, linking Eq. (1) and Eq. (2) without the need for an additional expression is possible as

" $ $ $ '

# % % % (

& !

#

!

! !

% +

!

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

(3)

"

#

!

&

" $ '

# % (

!

" + $

'

#$B

The finite element formulation should be derived by discretising the motion equation in Eq. (3). However, the equation is inconvenient for discretisation in this form because the term does not allow an analytical formulation. Therefore, it is replaced by a second-order polynomial as ( ) ( ) !" ! #$B ! " # +

!

"

"

#

" $

# % '

!

!

(4)

" # ! !

!

= +

+

!

"

$B& $B& &

" &

# % '

!

#

# + $

$B&

where a 0 , a 1 and a 2 are the polynomial coefficients. Following Newton interpolation polynomial given in general form (Kreyszig, 1993)

(

)

(

) (

( )

)

! " ! " !" # #"" #"" "" = + ! + ! ! ! #

(5)

(

)

(

)

the coefficients can be determined for a chosen range of the expression

as

! !

! " # $B& +

! !"!#!" ""#!#" ! ! " ! " " ! ! " ! " ! ! = " + = " " $

$

!

(6)

!

=

# #

with

! " # " # ! ! " ! ! ! ! ! # # # # # # ! # # # $ # # # & # !

" % '

! ! # # # #

! "

(7)

" " ! " ! " = $

$

"

=

=

#

! "

(

)

(

)

! # $ B&' E =

" " = +

# ) $ B&' !

$!"!%

(8)

=

! " !

! " !

!

!

By incorporating Eq. (4) into Eq. (3), the representation of the motion equation suitable for an analytical formulation of the finite element is obtained, in index notation, as

#

$ & & )

% ' ' *

!

$ & & )

% ' *

A

A

$

% ' *

! B&' B&' ! ! ! + & )

(9)

$ # " + = % &' E' ) &% " ' # + ( " " " " !!

A

" # ! ! +

!