**Nomenclature**


#### **Appendix A.**

*Appendix A.1. Summary of the Analysis Performed*

**Table A1.** Summary of the Analysis Performed.


*Appendix A.2. Obtaining the Natural Frequency*

*Step 1*: Calculate the fixed base natural frequency:

$$\mathbf{f\_{fb}} = \frac{1}{2\pi} \sqrt{\frac{3\mathbf{E I\_{T-f}}}{\left(0.243\mathbf{m}\_{\mathrm{eq}}\mathbf{h}\_{\mathrm{total}} + \mathbf{M}\_{\mathrm{RNA}}\right)\left(\mathbf{h}\_{\mathrm{total}}\right)^{3}}} = \frac{1}{2\pi} \sqrt{\frac{3 \times 1.635 \times 10^{12}}{\left[0.243(4200 \times 140) + 350000\right](140)^{3}}} = 0.303\,\mathrm{Hz}$$

This is also presentative of the natural frequency if the jacket is supported on deep embedded piles

*Step 2*: Calculate CJ for the stiffness of the springs:

Using the equations provided in Table 3, a preliminary estimate of the vertical stiffness for a rigid caisson foundation (with L/D = 1) is obtained as shown below;

f(υs) = -10υ<sup>3</sup> s−5.88υ<sup>2</sup> s -<sup>−</sup>0.34 ln <sup>L</sup> <sup>D</sup>+0.77+0.91υ<sup>s</sup> - <sup>−</sup>0.57 ln <sup>L</sup> D+0.6 +1 = -10(0.28) 3 −5.88(0.28) 2 (−0.34 ln(1)+0.77) +0.91(0.28)(−0.57 ln(1)+0.6)+1 = 0.97 KV <sup>=</sup> 2.31- L D 0.52 DESf(υs) = 2.31(1) 0.52(4)(40)(0.97)= 0.36 GN m KV1,2 <sup>=</sup> 0.36 <sup>×</sup> <sup>2</sup> <sup>=</sup> 0.72 GN m For α = 1, KR = KV1,2L<sup>2</sup> α 1 + α <sup>=</sup> 0.72 <sup>×</sup> <sup>12</sup><sup>2</sup> <sup>×</sup> 1 1 + 1 = 52 GNm <sup>τ</sup> <sup>=</sup> KRhtotal EIT−<sup>J</sup> <sup>=</sup> <sup>52</sup> <sup>×</sup> <sup>109</sup> <sup>×</sup> <sup>140</sup> 1.635 <sup>×</sup> <sup>1012</sup> <sup>=</sup> 4.45 CJ = τ <sup>τ</sup> <sup>+</sup> <sup>3</sup> <sup>=</sup> 4.45 4.45 <sup>+</sup> <sup>3</sup> <sup>=</sup> 0.77 f0 = CJ × ffb = 0.77 × 0.303 = 0.23Hz

Readers are refered to Jalbi and Bhattacharya [15] for the step-by-step derivation of EIT-J and KR. In essence, the vertical spring stiffness was factored by 2 as the method converts the 3D representation of the system into 2D. Therefore, each spring in Figure 11 is representative of two caisson foundations.

*Additional Step:* Bordón et al. [26] propose the following method to calculate group effects correction factors:

$$\text{Group Effect Factor} = \frac{1}{1 + 0.06(1 + 3.08 \text{N}) \left[ \left( 1 + 1.2 \text{(L/D)}^{0.53} \right) / \left( \text{s/D} \right) \right]}$$

where N is the number of foundations and s is the spacing. For the solved example, this results in reducing the vertical stiffness of the foundations by a factor of 0.63. This further reduces the natural frequency to 0.21 Hz after the repition of the calculation above.
