Page 224 - Pipelines and Risers

Installation Design 197

A structural analysis of this system requires the specification of the properties of the pipe
(stiffness, weight), the configuration of the stinger, any environmental loads and the contact
conditions on the seabed and the stinger. For the purpose of this discussion the dynamic
effects that arise from lay vessel motions, waves and the motion of the pipe will be neglected.

In order to predict the equilibrium configuration, the principle of virtual work is used
J,d& = Jtii ?dS + 6; . fdV (12.21)
dV
*
Customary symbols represent stress, strain and displacement. Here t represents the surface
tractions and f the body forces. Notice that the integrals are over the volume of the pipe and
over the surfaces of the pipe, both inner and outer.

If the deformation of the cross-section is neglected, the integrals simplify considerably and
they can be transformed into line integrals. We will assume this has been done. Let us
consider the different terms.

The body force in the pipe-lay problem is the weight of the pipe and its contents. The energy
related to this virtual work term is the potential energy, which is:
-Jz(jlg)ds (12.22)

where z represents the vertical coordinate of the center line, h the mass per unit length, g the
acceleration of gravity and ds the length of the line element along the center line.

The surface integral arises from the surface tractions, which are of diverse origins. The
pressure of the contents, the water pressure and the water-motion tractions on the wetted part
of the outer surface are the most obvious. Contact stresses arise on the seabed and on the
stinger. The pressure integrals are easily integrated and give rise to pressure terms in the
effective force and the buoyancy.

The contact surface tractions are important in connection with pipe twist. An example is
illustrative: suppose the touch-down point of the pipe on the seabed is forced sideways.
Friction forces arise due to the transverse displacement, and these forces will tend to twist the
pipe around the centerline because the outer radius acts as the lever of the force. The internal
virtual work must be evaluated taking into account the stress-strain history of each material
point, so that a correct plastic state is maintained. For pipelay, plastic flow is only allowed
over the stinger while elastic conditions are required in the underbend. We can, therefore,
formulate the strain energy as:
(12.23)

where the three constants represent section-factors and T = axial force, Mb = bending
moment and Mt = twisting moment.