Introduction to the P-delta effect
The purpose of a jack-up platform is to support a deck and mission equipment, e.g. a crane or derrick, above the water level. This mission equipment, together with the hull and variable deck load, makes up most of the weight of the platform. The weight is the P-part of the P-delta effect. The hull will move laterally due to environmental loads on the hull and legs. This lateral movement is the delta-part of the P-delta effect.
When the weight P is at an offset delta, this introduces an extra contribution to the overturning moment of the platform, in turn leading to another (smaller) contribution to the offset. Therefore, it is said that the P-delta effect is a secondary effect. In this definition, the extra offset will lead to another extra offset, ad infinitum. This is how some numerical solvers include geometric non-linearities such as the P-delta effect, requiring a number of iterations.
The P-delta effect as stiffness reduction of a beam
Another way of thinking about – and implementing of – the P-delta effect is as a stiffness matrix. This is introduced in E. Wilsons excellent book, http://www.edwilson.org/BOOK-Wilson/11-PDE~1.pdf .
As a simple example, think of a rod in tension. As most people are aware, a rod or rope in tension, has a restoring force against transverse displacement of its ends e.g. has a lateral stiffness. If a rod is in compression, this stiffness is negative, reflected in buckling behavior.
The forces Fi and Fj due to displacements vi and vj can be expressed as the following matrix equation:
If T is negative (compression P), the stiffness matrix is negative, thus giving a stiffness reduction.
Applying a negative stiffness of P/L on the hull is a simplified method proposed in ISO 19905-1, where P is the elevated weight and L the free leg length. However, the implicit assumption that the legs together behave like a rod may be an oversimplification.
In Calypso each leg is modeled as a stack of beams connecting intermediate nodes. A beam not only has end forces, but also end moments. If we assume that bending of the beam due to end-force is parabolic and cubic due to end-moment (valid for Euler beams – close enough for Timoshenko beams as used in Calypso), the following equation applies for the stiffness due to the P-delta effect on a beam:
Where Mi, Mj are end moments and φi and φj are end rotations. If T is set to be the axial load in the beam, this is a good reflection of the P-delta effect. This matrix can be added to the beam stiffness matrix, so that the system can be solved in one go. Based on the top-left item in the matrix, it can be concluded that if a simplified approach is used, a better negative spring stiffness would be 36T/30L.
The P-delta effect on jack-up platforms
Until this point in this article, the P-delta effect was considered for a single beam-column supporting a weight. A jack-up is best modelled as a bar-stool – a mass supported by a multitude of beam-columns. When a jack-up is subjected to a horizontal force, the following things happen:
- The hull will be offset from its initial position
- All legs will sustain a roughly equal bending moment
- The leeward leg will have an increased axial load whereas the windward leg will have a decreased axial load due to the overturning moment
So, for a jack-up leg, the P (axial load) is not just given by weight, but also by the effects from environmental actions.
The ISO standard for site-specific assessments of jack-ups (ISO 19905-1) distinguishes between a P-Δ effect (capital Delta) and a P-δ effect (small delta). The former pertains to the influence on overall jack-up behavior such as offset of the hull and overturning moment, whereas the latter considers the element level e.g. for the local chord beam-column check, which is outside of the scope of this article.
The overall behavior of a jack-up is not (much) influenced by the effects of variant in axial load between the legs. The stiffness of the leeward leg reduces, but the stiffness of the windward leg increases, netting to a similar overall horizontal stiffness of the platform as a whole.
However, the stiffness reduction of the leeward leg will lead to a moment redistribution to the stiffer windward leg. This effect is not captured in the definition in the ISO standard. It is sort-of intermediate between P-Delta (capital) and P-delta.
In order to capture the effect of axial load variation, one iteration is required. In the first run the stiffness matrices for the P-delta effect are set based upon weight alone and the environmental load is applied. In the iteration, the stiffness matrix due to the P-delta effect is adapted per element to account for the axial load found in the first run. Since the axial load is not further influenced by this, one iteration is sufficient.
The P-delta effect in dynamic simulations
Typically, time domain dynamic simulations of jack-up behavior are performed to calculate the dynamic amplification factor. This DAF can be determined based on overall platform behavior, for which the P-delta effect need not be implemented iteratively if a stiffness matrix approach is used. Furthermore, the foundation model for this type of simulation is typically a linear spring, the damping is applied as viscous damping and all other elements also have linear behavior.
From the facts stated above, it can be concluded that the jack-up model for dynamic simulation can be written as a linear model, if the P-delta effect is included as a stiffness matrix in each element, with P based on the platform and leg weight.
The wave forcing is non-gaussian, so a fully statistical frequency domain approach is not evident. However, the static and dynamic behavior being linear does open up possibilities to a more statistical approach of determining a DAF, based on a shorter time trace of wave forces.
The stiffness matrix implementation of the P-delta effect is the model of choice in Calypso. It allows for a simple and effective approach. This paper presents a comparison between several methods and shows that correctly implementing the P-delta effect can have a significant influence on the leg moments.