Modelling compression in flexibles

When a flexible line experiences compression, it responds by deflecting transversely: the magnitude of the deflection is controlled by bend stiffness. Under static conditions, the behaviour of an initially straight section of line under pure axial loading is described by classic Euler buckling theory. This defines the maximum compressive load – the Euler load – which a particular length of line can withstand before transverse deflection occurs. The Euler load is a function of the length of the straight section, the bend stiffness and the end conditions. For a simple stick of length $l$, bend stiffness $EI$, with pin joints at each end, the Euler load is $\pi^2 EI / l^2$. The Euler load is derived from a stability analysis: it tells us the value of axial load at which transverse deflection will occur but nothing about the post-buckling behaviour.

Under dynamic loading conditions, the transverse deflection is resisted by a combination of inertia and bending. OrcaFlex is fully capable of modelling this behaviour provided the discretisation of the model is sufficient, i.e. provided the segments are short enough to model the deflected shape properly. Another way of saying the same thing is that the compressive load in any segment of the line should never exceed the Euler load for the segment.

Why are these two statements equivalent? Imagine the real line replaced by a series of rigid sticks connected by rotational springs at the joints – this is essentially how OrcaFlex models the line. Under compression, the line deflects: the sticks remain straight and the joints rotate. Provided the wavelength of the deflection is longer than the length of the individual sticks then the rigid stick model can approximate it: shorter sticks give a better approximation.

If the compressive load reaches the Euler load for an individual stick, then the real line which the stick represents will start to deform at a shorter wavelength, and deflections within the stick length become significant. Clearly, this stick model is no longer adequate. By replacing each long stick by several short ones, we can make the Euler load for each stick greater than the applied compressive load. Each stick still remains straight, but we now have more sticks with which to model the deflected shape.

This gives us a convenient way of checking the adequacy of our model: provided the compressive load in each segment always remains less than the Euler load for that segment, then we can have confidence that the behaviour of the line in compression is adequately modelled. OrcaFlex makes this comparison automatically for all segments and reports any infringements in the statistics tables. The segment Euler load is also drawn on tension range graphs (as a negative value – compression is negative) so that infringements are clearly visible.

If the segment Euler load is infringed during a simulation, then we have to decide what to do about it. If infringement occurs only during the build-up period, perhaps as a result of a starting transient, then we can safely ignore it. If it occurs during the main part of the simulation, then we should examine the time histories of tension in the affected areas. Where infringements are severe and repeated or of long duration the analysis should be repeated with shorter segments in the affected area. However it may be acceptable to disregard occasional minor infringements of short duration on the grounds that:

• Transverse deflection caused by compression takes some time to occur because of inertia.
• The segment Euler load used in OrcaFlex as a basis for comparison is the lowest of the various alternatives, and assumes pinned joints with no bend stiffness at each end of the segment. This is a conservative assumption.

Whether or not to disregard an infringement is a decision which can only be taken by the analyst in the context of the task in hand.