Multiple non-linear regression analysis for the stability of non-overtopping perforated quarter circle breakwater
Binumol S1, 3 , Subba Rao2 and Arkal Vittal Hegde2
1Department of Civil Engineering, Shinas College of technology, Sultanate of Oman & formerly Research Scholar, Department of Applied Mechanics & Hydraulics, National Institute of Technology Karnataka, Surathkal, Mangalore, India.
2 Department of Applied Mechanics & Hydraulics, National Institute of Technology Karnataka, Surathkal, Mangalore, India.
3 [email protected]
Abstract. Breakwaters have been built throughout the centuries for the coastal protection and the port development but their structural development as well as their design procedure is under massive change. New ideas and developments are in the process of being tested regarding breakwater layout for reducing wave loads and failures. Breakwater design is increasingly influenced by environmental, social and aesthetical aspects and new types of structures are being proposed and built. Quarter circle breakwater (QBW) is a new type evolved is found to possess merits of caisson as well as perforated breakwaters such as low weight, requires less materials, suited for poor soil conditions, easily transported, handled and placed at the site, aesthetically pleasing, cost effective, eco- friendly and stable. In the present study, an attempt has been made to study the stability of seaside perforated QBW with varying perforations using physical models. The results were plotted as non-dimensional graphs and it was observed that the stability parameter decreases with increase in wave steepness for all values of ratio of water depth to height of breakwater. Multiple non –linear regression analysis was done and equation for the best fit curve with a higher regression coefficient was obtained by using Excel statistical software – XLSTAT.
Ports are considered to be the most important transit locations to carry out the world trade through seaways, which need to be protected from the disturbances due to the incoming waves. A seaport plays an important role in the sector of sea transportations, exports, imports, tourism, and travel, and thus is an important ingredient of economic growth. Water transport is a major economy input for a nation with over 82% of world trade in tons and 94% of world trade in tons-kilometres which are moved by shipping and thereby through ports.

The basic requirement of any port, harbor or marina is a sheltered area free from the waves. In the coastal areas where natural protection from waves is not available, the development of harbour requires an artificial protection for the creation of calm areas. For harbours, where perfect tranquility conditions are required, large structures such as rubble mound breakwaters or vertical wall breakwaters are used. Most of the breakwaters are used to create tranquil conditions in the lagoon and at the entrance channel of ports, for maneuvering of ships and port operations. Mostly breakwaters are also used as berthing structures along with protecting the harbor area. Sometimes they are used to protect beaches from erosion due to the destructive wave forces.

Due to fast growing need of the universe and advances in technology different types of composite breakwater are being developed. Semicircular breakwater is a composite structure first developed in Japan at the beginning of the Nineties and first adopted for the formation of the harbor in Miyazaki Port, Japan. Quarter circle breakwater (QBW) is a new-type breakwater first proposed by Xie et al. (2006) on the basis of semicircle breakwater. The present study investigates the stability of an emerged seaside perforated quarter circle breakwater with varying percentage of perforations, for different water depths and wave conditions. The major objectives include the study of the effect of incident wave steepness, S/D (spacing between perforations/diameter of perforations) ratio or percentage of perforations and water depth on the stability characteristics of perforated QBW.
Experimental Investigations
The experiments were conducted in a two dimensional wave flume available in the Marine Structures Laboratory of Department of Applied Mechanics and Hydraulics, National institute of technology Karnataka, Mangalore, India.
Wave Flume and Instrumentation
The physical model study for regular waves was conducted in a two dimensional wave flume 50 m long, 0.71 m wide and 1.1 m deep (Refer Figure 1). Waves of height ranging from 0.08 m to 0.24 m heights and periods from 0.8 sec to 4.0 sec can be generated with this facility.

Figure 1. Longitudinal Section of Wave Flume.
Test Model
The breakwater model basically consists of two parts: a top quarter circular shaped caisson and a bottom base slab made of precast concrete founded on rubble mound base. The caisson was made with Galvanized Iron (GI) sheet of 2 mm thickness. To simulate the field conditions of wave height, wave period and diameter of perforation by application of Froude’s law, a geometrically similar model scale of 1:30 has been selected for the present experimental investigations.

Figure 2. Typical Cross section of Perforated QBW.
For present study, QBW of 0.55 m radius was selected to satisfy the condition that the model will be emerged at all water depths and remains non-overtopped under all wave conditions considered for the study. The QBW model was fixed to the base slab with the help of stiffeners. The perforations are provided in the quarter circular front portion of the breakwater model, by drilling the holes. Figure 2 shows the typical view of quarter circle breakwater considered for the study.
The model after casting was placed over the rubble mound foundation of thickness 0.05 m using stones weighing from 50 to 100 g having a slope 1:2. The QBW model together with rubble mound foundation was usually placed 30 m away from the wave flap. Three probe method proposed by the researchers 4 was used for measuring the incident and reflected wave height. The first probe was placed L distance from the centre of the probe, and the distance between the other probes were equal to L/3, where L is the wave length. A burst of five waves was generated to avoid the successive reflection. The surface elevation measured by the probes were recorded by the wave recorder and the voltage signals are converted into wave heights and wave period by using the lab wave recorder software provided by EMCON (Environmental Measurements and Controls), Kochi, India. The range of parameters selected for the studies are mentioned in Table 1.

Table 1. Range of Experimental variables.
Parameters Experimental Range
Wave Specific Parameters
Incident wave height, Hi
Water depth, d
Time period, T
Structure specific parameters
Radius of the structure
S/D ratio
Diameter of perforation
0.03, 0.06, 0.09, 0.12, 0.15, 0.18 m
0.35, 0.40, 0.45 m
1.2, 1.4, 1.6, 1.8, 2.0, 2.2 sec

2, 2.5, 3, 4, 5
0.016m, 0.020m
The present experimental investigations are carried out with the following test conditions:
The sea bed is rigid and horizontal and it is assumed that the sediment movement does not interfere with the wave motion and do not affect the model performance.
The waves are periodic and monochromatic.
Wave reflection from the structure does not interfere with freshly generated incident waves, since the waves are generated in bursts.
Secondary waves generated during the test are not considered.
Wave reflection from the flume bottom or flume side walls is not considered.
The density difference between freshwater and seawater is not considered.
Bottom frictional effects have not been accounted.
Results and discussions
Studies were conducted on emerged seaside perforated QBW with different spacing to diameter (S/D) ratios under different water depths (say 0.35, 0.40 and 0.45 m) and varying wave conditions. Variations of minimum weight required for sliding stability with different wave specific and structural specific parameters were studied, and the variations were then plotted graphically using non-dimensional parameters. The stability based on sliding performance was represented by a non-dimensional stability parameter (W/?Hi2), where W is the minimum weight of the QBW required to resist the sliding per unit length of the breakwater, ? is the specific weight of water and Hi is the incident wave height.
4.1 Influence of incident wave steepness on W/?Hi2
The variation of W/?Hi2 with Hi/gT2 at different water depths for QBW radius 0.55 m and S/D ratio equal to 2, 2.5,3,4 and 5 were plotted as shown in Figure 3. It was clear from the graphs that W/?Hi2 decreases with increase in Hi/gT2 for all values of d/hs.

For S/D = 2 (b) For S/D= 2.5

For S/D = 3 (d) For S/D= 4

(e) For S/D= 5
Figure 3. Influence of Hi/gT2 on W/?Hi2 for S/D= 2,2.5,3,4 and 5 and at different water depths.
For all models, it was observed that when the incident wave steepness increases, the dimensionless stability parameter decreases. This is because for a given wave height and long period waves (low steepness) exert more force on the caisson demanding more minimum weight and short period (steep) waves transfer less force, hence low minimum weight. The sliding due to increase in wave force is overcome by increasing the weight of breakwater by adding additional weight into the caisson. The variation of W/?Hi2 with wave steepness for different water depths and S/D ratio for seaside perforated QBW were summarized in table 2.
Table 2 Variation of W/?Hi2 with Hi/gT2
S/D ratio Water depth in cm d/ hs Variation of W/?Hi2 with Hi/gT2
2 0.45
0.35 0.732
0.569 3.335 – 10.532
3.236 – 8.766
2.225 – 7.249
2.5 0.45
0.35 0.732
0.569 3.198 – 10.269
3.155 – 8.546
2.110 – 6.967
3 0.45
0.35 0.732
0.569 3.566 – 11.223
3.385– 9.566
2.472 – 7.989
4 0.45
0.35 0.732
0.569 3.882 – 11.668
3.555 – 9.711
2.729 – 8.412
5 0.45
0.35 0.732
0.569 4.056 – 12.850
3.779 – 10.695
3.006 – 9.355

4.2 Influence of water depth on W/?Hi2
From the results, it is clear that as the water depth increases, the value of W/?Hi2 increases and the minimum value for W/?Hi2 are observed at a water depth of 0.35 m. As water depth increases from 0.35 m to 0.40 m there is an increase in W/?Hi2 by 17.30 % to 31.24%. When water depth increases from 0.35 m to 0.45m there is an increase in W/?Hi2 by 31.17 % to 33.28%. Therefore it is clear that the structure is safe against sliding with minimum weight including additional weight at lower water depths. This is because higher the water depth, greater is the area of the QBW model structure exposed to wave action, and hence, the increase in d/hs imparts more force and therefore increase in W/?Hi2. This means that for a constant S/D ratio more structure weight is required for sliding stability in larger depths compared to smaller water depth.

4.3 Influence of S/D ratio on W/?Hi2
From the results for W/?Hi2 for varying ranges of Hi/gT2, it is observed that W/?Hi2 increases with increase in S/D ratio. At a water depth equal to 0.40 m, the percentage reduction in W/?Hi2 for S/D = 4 , 3, 2.5 and 2 are 5.92% to 9.20% , 10.42% to 10.55% , 16.51% to 20.09% and 14.36% to 18.04% with respect to S/D = 5. At a water depth equal to 0.45 m, the percentage reduction in W/?Hi2 for S/D = 4 is 4.29% to 9.19% compared to S/D = 5. The percentage reduction in W/?Hi2 for S/D = 3, 2.5 and 2 are 12.32% to 12.66%, 20.08% to 21.15% and 17.77% to 18.03% with respect to S/D = 5.For lower values of S/D, W/?Hi2 is found to very less compared to higher values except in the case of S/D ratio equal to 2. For lower values of S/D perforations encountered are more resulting in dissipating of major portion of the wave energy and hence force exerted on the QBW caisson will be very less. Therefore the weight required for resisting sliding stability will be very less and resulting in lower values for W/?Hi2.

4.4 Equations developed for stability parameter
The results for the experimental studies on stability characteristics for impermeable QBW for different breakwater radius at different water depths and wave conditions are combined into suitable dimensionless terms. The regression analysis is done by using Excel statistical software – XLSTAT and the equation for the best fit curve is obtained.
The equation for W/?Hi2 for seaside perforated QBW was derived as follows:
“W/?Hi2 = 6.813(Hi/gT2)-0.503″+”31.580(d/hs)8.357 “+”0.0008(d/hs)4.524 “-0.0″841”
Figure 4 shows the comparison between the measured and predicted values of stability parameter W/?Hi2 for perforated QBW.

Figure 3. Comparison between measured and predicted values of W/?Hi2 for perforated QBW.
Based on the experimental results of model studies on seaside perforated QBW, following conclusions are drawn:
For all values of d/hs and S/D ratio, W/?Hi2 decreases with increase in Hi/gT2. The minimum values for W/?Hi2 for QBW of radius 0.55 m and with S/D= 2.5 are 2.110 for Hi/gT2= 6.241 x10-3 and at 0.35 m water depth.
For various S/D, W/?Hi2 increases with increase in water depth for all ranges of Hi/gT2. The percentage increase in W/?Hi2 for a water depth equal to 0.45 m compared to 0.35 m varies from 32.15% to 34.02% for QBW of radius 0.55 m with S/D= 2.5.
The stability parameter W/?Hi2 decreases with decrease in S/D ratio for all values of Hi/gT2 and d/hs. But a reverese trend is observed in the case of seaside perforated QBW with S/D= 2. When S/D is reduced there will be more dissipation of wave energy and QBW is subjected to less wave force resulting in less weight of QBW to prevent sliding or lesser values of W/?Hi2. Further decreasing S/D to 2 causes back propagation of wave and hence exert more force on QBW resulting in higher values for W/?Hi2.
Uncertainty analysis was done for the various tests conducted and the equation for the best fit curve with a good regression coefficient (R2 value) was found out.
From the experiments conducted and the comparative analysis, it is clear that the perforated QBW of S/D = 2.5 has better performance characteristics and can be adopted in the site with favourable condition.

The authors are thankful to Director, NITK Surathkal and the Head of Applied Mechanics Department, NITK Surathkal for all the support and encouragement in the preparation of this paper.
S L Xie, Y B Li, Y Q Wu and H B Gu 2006 Preliminary research on wave forces on quarter circular breakwater Ocean Engineering journal 24(1) L14
S Binumol, Subba Rao and Arkal Vittal Hegde 2017 Wave Reflection and Loss Characteristics of an Emerged Quarter Circle Breakwater with Varying Seaside Perforations, Journal of The Institution of Engineers (India): Series A Springer,98 (3) L311
Hughes S A 1993 Physical models and Laboratory techniques in Coastal Engineering Advanced series on Ocean Engineering, World Scientific Singapore.
Issacson M 1991 Measurement of regular wave reflection Journal of Waterway, Port, Coastal and Ocean Engineering, 117(6) L553
Jiang X L, Gu H B and Li Y B 2008 Numerical simulation on hydraulic performances of quarter circular breakwater China Ocean Engineering 22(4) L 585
Luwen Qie, Xiang Zhang, Xuelian Jiang and Yinan Qin.2013Research on partial coefficients for design of quarter-circular caisson breakwater. J. Marine Sci. Appl 12 L 65
Misra S C 2001 Uncertainity analysis in hydrodynamic tests. Proceedings, International Conference in Ocean Engineering, L207
SHI Yan-Jiao, Mi-ling W U, Jiang X and Li Yan-bao 2011 Experimental researches on reflective and transmitting performances of quarter circular breakwater under regular and irregular waves China Ocean Engineering 25 L469
Xie S L, Li YB , Wu Y Q, and Gu H B 2006 Preliminary research on wave forces on quarter circular breakwater Ocean Engineering, 24 L14