Cyclic stiffness degradation in non-linear jackup dynamics
Authors: Dean Edward T.R. , Metters Rich 
Published: July 2009

Abstract
SNAME (2002) Guidelines for Site Specific Assessment of Mobile Jack-Up Units gives equations to describe degradation of foundation stiffness for jackup spudcans, depending on the size of a load cycle within a yield envelope. Using these and other proposals, this paper develops a simplified dynamic analysis of jackup cyclic response. Under static conditions, a lower bound collapse load for a pinned foundation can be larger than for an encastré assumption. Under dynamic conditions, important non-linearity due to stiffness degradation is confirmed. Chaotic dynamic responses may be possible.

Introduction
The load-deflection response of the elevated hull of an installed jackup, subject to environmental loading, and the stresses at the leg-hull connections, depend on the stiffness of the spudcan foundations, and so on the moduli of the soils supporting them (Osborne et al, 1991; Dean et al 1992; Temperton et al, 1997; Nelson et al, 2000; Cassidy et al, 2002; Dier and Carroll, 2004). Typically, soil moduli decrease with increasing cyclic strain amplitude (Atkinson, 2000). Soil stiffness can also change as a result of cyclic strain history, and due to buildup of excess pore water pressures, both due to a sequence of storms, and during a design storm (Hsu, 1998).
SNAME (2002) uses the concept of a yield envelope to determine combinations of vertical and horizontal loads and moments on a spudcan that cause significant elasto-plastic responses. The envelope is used as if it was a failure envelope in the Step 2 Bearing Capacity Check, which includes an option to allow for rotational foundation stiffness. If the jackup fails Step 2, it can still be accepted if it passes a Step 3 Displacement Check, which can include allowance for hardening. Sophisticated models of the yield envelope approach include Schotman (1989), Dean et al (1997a, b), Van Langen et al (1997), Cassidy et al (2004), Bienen et al (2006), and others.
This paper explores the guidelines using a simplified jackup response model. Results are found to be potentially conservative if stiffness degradation is ignored. Stiffness degradation is shown to produce significant non-linearity in dynamic responses. An implication appears to be that observations in mild seastates do not extrapolate linearly to more severe seastates.

Simplified jackup model
Figure 1a shows a simplified conceptual model of a 3-legged jackup. The single, bow leg is on the left, and is identified with subscript 1. The two aft legs are on the right, subscript 2. The legs have lengths L.
A horizontal environmental load H, due to wind, wave, and current, is applied to the jackup at some height ?L above the level of the bearing areas of the spudcans. It induces a horizontal reaction H1 at the bow spudcan, and reactions H2 at each of the aft spudcans. If the masses of the legs and spudcans are ignored compared to the mass Mh of the hull, if the hull is rigid, and if some viscous damping is assumed related to the hull motion, then application of Newton's Laws of Motion gives the first equation in Figure 0H1b, where y is the horizontal displacement of the hull at some time t, and C is viscous parameter.
The present paper assumes the net reaction Hi at spudcan " i?? is applied to the leg by the rigid hull. Let yi be the horizontal displacement rightwards at the spudcan, and let Mi be the resisting moment from the soil. Let ? be the rotation of the rigid hull, positive clockwise, and let ?i be rotation of the spudcan. Assuming the leg acts as a cantilever encastré at the hull, elastic beam bending theory gives the second and third equations in Figure 1H1b.