Formula determination

The evaluation of data, described in subsection 5.3.5, proved that results obtained by the author were an accurate representation of the response characteristics expected during the stability analysis. With the credibility of the results verified, a design approach can be implemented to determine a proposed formula for the scope of the study.

The results of Brebner & Donnelly (1962) proved to serve as a valuable guide for the work conducted by the author. With a factor of H1/10 applied to their data, the graph in Figure 58 proved to resemble similar results to the experimental tests. However, the fact that their tests were conducted for regular waves makes their results not as comparable as the results of the other methods stated in the literature. Therefore, the work of Brebner and Donnelly is not used to aid in the development of a new formula.

Although Tanimoto et al. (1982) used 3.5% damage as their critical stability boundary, the applicability of their results has been rather successful in practice. The results of their formula had a distinct correlation with the experimental test, even though the influence of the relative foundation depth is not accounted for in their work. Therefore, it was essential to consider their work for the development of a new formula.

The work of Madrigal and Valdes (1995) proved to have the most significant resemblance towards the results of the experimental tests. The assumed values outside of the range of [0.5 -0.8], determined by their formula, is also taken into consideration for the development of the new formula.

Page | 79 With the results of the experimental tests proven accurate, an approximated average curve could be determined from the experimental data. The average curve, similar to the methods used in the literature, is drawn from the results of the critical stability number against the relative foundation depth for each of the three wave periods. The average curve thus represents the values needed for all three of the wave periods combined, as illustrated in Figure 59.

The average graph, suggested by the author, includes parameters that are susceptible to wave period variation, thus a bandwidth of the standard deviation is applied to the average graph to account for the error. The results, depicted in Figure 59, would serve as the foundational base for the determination of a new formula.

Figure 59: Test results proposed average graph

The proposed graph (“Author Average”), drawn from the critical averages of the experimental work, proved to be comparable to the present formulas used in practice. The work of Madrigal and Valdes (1995) proved to have the most significant resemblance towards the “Author Average” graph, as seen in Figure 59.

As expected (subsection 5.3.5), the results of Tanimoto et al. proved to have a higher estimated stability number for the small relative foundation depths. This method does, however, resemble similar projected stability numbers, in correlation towards the author’s work, as the relative foundation depth increases.

0 0.5 1 1.5 2 2.5 3 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Stab ili ty N u mb er ( N S )

Relative foundation depth ( d1/d )

Page | 80 By evaluating the literature, it was clear that the present formulas available in practice lack information regarding small relative foundation depths (d1/d) ranging from [0.35 – 0.55]. The work of Tanimoto et al. (ranging from 0.3 ≤ d1/d ≤ 0.9), could as a first approximation, be regarded as valuable towards the study, even though their test results were based on deeper relative depths (d/L). The work of Madrigal and Valdes were applied in relative depths similar to that of the author’s but only applies to relative foundation depths between [0.5 -0.8].

With all this information taken into consideration, an approach towards a new formula is proposed for relative foundation depths between *0.35 ≤ d1/d ≤ 0.55+. All the data points, acquired from the graph in Figure 59, are considered for the determination of a new proposed formula. The results of the author, together with the results determined from the literature, were plotted in a scatter, in order to estimate a suited trendline through the data points, as depicted in Figure 60. All the data points available in these ranges are illustrated as ‘Series1’.

By analysis, the linear trend line served as the best fit for the data at hand. It should be noted that the second order polynomial trend line had a correlation that was slightly higher (0.68) than the linear trend line, however, the linear trendline was chosen for its simplicity with a minimal difference in the correlation of the data. The proposed trend line, as seen in Figure 60, is accompanied by a 95% confidence band. The confidence band is used to evaluate the credibility of the proposed trend line. The determination of the 95% confidence band is stated in Appendix J.

Page | 81 Through evaluation, the data points under consideration are well represented by the linear curve of the trend line. As expected, the trend line underestimates the stability values for the region of [0.35 ≤ d1/d ≤ 0.45+ in relation to the work of Tanimoto et al.

The proposed trend line would enable the author to acquire a formula that would express the minimum stability number needed for the associated relative foundation depth. Similar to the work of Madrigal and Valdes (1995), the author applied a linear trend line to the data values. The proposed formula determined by the author is thus:

( )

Valid for: Irregular head-on waves

Toe berm formed by two layers of quarry stone Relative foundation depth (d1/d) = [0.35 – 0.55]

∆ = 1.65