A Flexelement Analysis
Authors: Hannus K.G. 
Published: October 2008

ABSTRACT
The flexelement (or flexjoint) is a laminated elastomer/steel structure, which is used to give the TLP tethers and marine risers their cocking capability. Due to complex geometrical and material behaviour, the flexelement design has been relying heavily on testing both during development and at verification.
The work presented herein describes a full non-linear analysis of a flexelement at combined tension and bending load. The objective of the work has been to determine the stress distribution in the steel laminates. The analysis results are verified through strain gauge measurements of edge strains in tests of full scale components. The agreement is very good for the compressive stress edge, while the tensile stresses are somewhat overpredicted by the analysis.
The performed analysis provides new information for better understanding and prediction of flexelement behaviour.

INTRODUCTION
The flexelement is the component that permits large offsets of a TLP without overstressing the tethers or the risers. It is a multilayered steel/elastomer structure. Thin elastomer layers are sandwiched between steel laminates of spherical shape (also referred to as "reinforcements", "shims" or "interleaves"). The elastomer squeezing is restrained by the bonding to the steel laminates, which give a high axial stiffness while maintaining a low cocking stiffness. A schematic illustration of a typical flexelement, in between a shaft and a housing, is given in figure 1. Shaft tilting is marked by dashed lines.
The initial design of a flexelement is typically based on analytical closed form solutions of the behaviour of an elastomer layer under load. The solution may either be based on plane strain, the "shape factor" approach, ref. /1/, or polar coordinates, ref. /2/. This simplified method gives a fairly accurate prediction of the stiffness characteristics (for axial stiffness, the compressibility should be considered). An initial estimate of the elastomer strains can also be derived, but the steel laminate design relies entirely on experience data.
This stage is then followed by prototype building and testing, some design iterations may be required and at the end "the component works but nobody knows exactly why", ref. /3/.
In the design of related steel components, finite element analyses are used extensively, ref. /4/, however the material and geometrical non-linearities of the multilayered flexelement implies that the size of the analysis well exceeds what is common within structural design. Therefore, the analyses presented in literature so far have been limited to axi-symmetric models (in linear analysis, the tilting may be considered through Fourier decomposition). That is not satisfactory as illustrated by the work reported here, typically only one third of the steel laminate hoop strain/stress is caused by the axial load, the rest is induced by the tilting.
To clarify any uncertainties in the design and to add to the physical understanding of the flexelement behaviour, a verication analysis of a tether system flexelement was instigated. The analysis was to be nonlinear and include bending (angular) load, which requires a three dimensional model of the full flexelement (180 degree sector).