## Isolating Heave/Pitch Effects on Vertical Velocity at the Spudcans while Going on Location

###### Jul 8, 2015 News Archive

In the following paragraphs, we make a few observations regarding some of our findings and theories on permissible wave height curves for jackups and liftboats. Much of the below discussion is based on going on location analyses where roll is zero (i.e., head or stern seas). This makes the discussion simpler, but it does not imply that there aren’t rotations about other axes – “pitch” is solely used for simplicity.

As mentioned in **Part 2** of our Self-Elevating Unit series article, the permissible wave height curves for a given jackup are a function of water depth and soil stiffness, but they tend to follow a general shape. From the various jackups we have analyzed, we have found that the lowest permissible wave height tends to occur at periods that are 2-5 seconds lower than the pitch natural period. From these results and by looking at both the heave and pitch motion RAOS of a jackup, we came to describe the expected shape of the permissible wave height curve as follows:

- At smaller periods, as both heave and pitch motions are small, the permissible wave heights are (relatively) high. Then,
- As both pitch and heave motions increase for the intermediate periods, the permissible wave height curve reaches its lowest value, but since velocity reduces with period, the lowest permissible wave height typically occurs at periods lower than the pitch natural period. Then,
- For the larger periods (greater than the pitch natural period), as pitch decreases and heave remains constant, the permissible wave height curve starts to increase again. For these larger periods, the increase is slight, and eventually the curve tends to flatten out (i.e., the curve will change its concavity and tend towards a set limit).

**Figure 1 – Variation of Vz and its rate of change for uniform heave**

As a means to test the above assertion, we took one of the models previously analyzed for periods between 6 and 16 seconds and extended the range to 26 seconds (See Figure 2). Since the goal is simply to view the shape of the permissible wave height curve as it pertains to its behavior in the 3 identified period segments (small periods, intermediate periods and large periods), and in order to protect the confidentiality of the work we have done for our clients, the curves are presented with the vertical scale blanked out and with no details on the jackups, water depth or soil conditions. It is noted that in our simplified analysis methodology for establishing the permissible wave height, we limit the wave heights to multiples of 0.5ft. As such, the curves are not very smooth.

**Figure 2 – Permissible Wave Height Curve (going well past Pitch Tn)**

As a matter of interest, we note that the traditional way to determine going on location capabilities is based on conservation of energy, as described in DNV Classification Notes Note. 31.5 – Strength Analysis of Main Structures of Self-Elevating Units (February 1992). This approach, while conservative in some ways, is based solely on rotation and ignores heave. In that formulation, the vertical and horizontal loads on the leg due to impact are proportional to the product of pitch (or roll) angle and frequency. In other words, they are linearly proportional to the rotational velocity of the jackup. What that means is that a permissible wave height curve would look like the inverse of the pitch velocity RAOs. Therefore, the curve will increase with a concave-up shape indefinitely as the periods get larger and larger, since both pitch motion and frequency (therefore pitch velocity) decrease as the periods get larger and larger.

Reflecting on the energy approach used in the DNV Classification Notes Note 31.5, it is evident that the impact loads are related to velocity. From our perspective, the impact loads are not just affected by rotational velocity, but by spudcan velocity, as influenced by both pitch and heave. With this in mind, Figure 3 shows an RAO plot of the vertical velocity at the FORWARD spudcan for a generic jackup. The figure shows the heave and pitch

**Figure 3 – FWD Spudcan Vertical Velocity due to Heave and Pitch**

**Figure 4 – Phase Angles for Pitch/Heave-induced Vz of FWD Can**

As can be seen in Figure 4, the steepest changes in phase angle take place near the natural periods (~10sec for heave and ~16 sec for pitch in this case). This is consistent with what we know from single-degree-of-freedom response to harmonic excitation theory, which states that the phase angle is given below.

From the expression for phase angle, a 180-degree phase angle shift is expected at the natural period when there is zero damping (i.e., there is a discontinuity in the phase angle curve). For cases with non-zero damping, the rate of change in phase angle for frequency ratios in the vicinity of r=1 decreases with damping. From Figure 5, the phase angle curves for the heave and pitch contributions to the vertical velocity of the FWD spudcan are consistent with the 1-DOF theory.

**Figure 5 – Phase Angles in the vicinity of Tn for Different Damping**