A Study on Stability of Floating Architecture and Its Design Methodology

Abstract

Herein, the authors describe an overall approach to the architectural design of floating structures such as floating houses. The primary aim of this study is not to present a method for stabilizing floating structures, but rather to provide a design synthesis method for use when designing such structures. More specifically, we propose an integrated procedure for use at the preliminary design stage of such structures that systematically facilitates their overall design. As an inclining platform could endanger the people on board, it is necessary to determine an adequate metacentric height in order to prevent such occurrences. This measurement, which is defined as the distance between the center of gravity of a floating structure and its metacenter, quantifies the initial static stability of a floating body. Based on this idea, we consider the associated problems as well as the methods used in practical procedures, and combine them to introduce a unique approach called the “required GM” method. We also discuss the different and various aspects used in basic configuration determinations of floating architectural structures, such as the aspect of static stability and the overall process used at the conceptual design stage. In addition, illustrative examples of an idealized floating platform embodying the simplest possible structures are provided to illustrate these points.

Appendices

Appendix A

In general, steady wind force $\left( {\Delta {\text{F}}_{\text{Z}} } \right)$ at z m can be obtained by the following equation:

$$\Delta {\text{F}}_{Z} = {\text{P}}_{\text{Z}} \cdot {\text{A}} = \frac{1}{2}{\rho } \cdot {\text{U}}_{\text{Z}}^{2} \cdot {\text{C}}_{\text{D}} \cdot {\text{A}}$$

(20)

where P_Z is the wind pressure (kg/m²) at z m, C_D is a drag coefficient, A is the projected area (m²), and ρ is the air density.

In general, accurate C_D values are obtained via wind tunnel tests.

Here, it should be note that wind velocity is measured at the height of 10 m and the average value over a period of 10 min is used. The wind velocity (U_Z) changes along the vertical location are shown in Fig. 18, and can be estimated by the following equation. Note that wind velocity is lower near the ground due to friction.

$${\text{U}}_{\text{Z}} = {\text{U}}_{10} \left( {\frac{\text{z}}{10}} \right)^{\alpha }$$

(21)

where U₁₀ is the wind velocity at a height of 10 m, and α is the surface roughness.

Fig. 18

Schematic view of wind speed

It is known that the value of α is 1/7 on the sea surface and 1/4 in an urban area.

Appendix B

According to Ref. [6], the static righting lever GZ at an inclining angle φ of a wall sided vessel is expressed as follows:

$$\overline{\text{GZ}} \left( \phi \right) = \sin \phi \left( {\overline{\text{GM}} + \frac{1}{2}\overline{\text{BM}} \tan^{2} \phi } \right)$$

(22)

Supposing an overturning moment due to the sum of various components $\left( {\sum\nolimits_{\text{i}} {{\text{M}}_{\text{i}} } } \right)$ and the righting moment (M_R), the following expression is established:

$$\overline{{{\text{GZ}}}} \left( \phi \right) \cdot {\text{W}} = {\text{M}}_{{\text{R}}} \left( \phi \right) = \sum\limits_{{\text{i}}} {{\text{M}}_{{\text{i}}} } \left( \phi \right)$$

(23)

Equation (22) is then multiplied by the weight of a floating foundation (W) to give

$${\text{W}} \cdot \overline{\text{GZ}} \left( \phi \right) = \sum\limits_{\text{i}} {{\text{M}}_{\text{i}} \left( \phi \right) = {\text{W}} \cdot \sin \phi \left( {\overline{\text{GM}} + \frac{1}{2}\overline{\text{BM}} \tan^{2} \phi } \right)}$$

(24)

After rearranging, we have

$${\text{W}} \cdot \sin \phi \cdot \overline{{{\text{GM}}}} = \sum\limits_{{\text{i}}} {{\text{M}}_{{\text{i}}} } \left( \phi \right) - \frac{1}{2}{\text{W}} \cdot \sin \phi \cdot \overline{{{\text{BM}}}} \cdot \tan ^{2} \phi$$

(25)

Dividing Eq. (25) by ${\text{W}} \cdot \sin \phi$ on both sides gives

$$\overline{\text{GM}} \left( \phi \right) = \frac{{\sum\nolimits_{\text{i}} {{\text{M}}_{\text{i}} \left( \phi \right)} }}{{{\text{W}} \cdot \sin \phi }} - \frac{1}{2}\overline{\text{BM}} \times \tan^{2} \phi .$$

(26)

When $\phi$ is small, the second term of Eq. (26) can be ignored. Thus, the value of the $\overline{\text{GM}}$ which is required (Req. GM) for a small inclination is given by the following equation:

$$\begin{aligned} {\text{Req. GM}}\;\left( \phi \right) & = \frac{{\sum\nolimits_{{\text{i}}} {{\text{M}}_{{\text{i}}} } }}{{{\text{W}} \cdot \sin \phi }} \\ & = \frac{{180.}}{{\pi \cdot \Phi {\text{d}}}} \times \frac{{\sum\nolimits_{{\text{i}}} {{\text{M}}_{{\text{i}}} } }}{{\text{W}}} \\ \end{aligned}$$

(27)

or

$$\Phi {\text{d}}\;\left( {{\text{in degrees}}} \right) = \frac{{180.}}{{\pi \cdot {\text{GM}}}} \times \frac{{\sum\nolimits_{{\text{i}}} {{\text{M}}_{{\text{i}}} } }}{{\text{W}}}$$

(28)

where $\Phi {\text{d}}$ is the inclining angle in degrees.

Thus, a “required GM” method that evaluates and compensates for various heeling moments under a variety of combined environmental loadings is proposed.