In the performance-based design of a breakwater armor layer, it is often necessary to calculate its damage accumulated over the life cycle of the breakwater. Two methods for calculating the cumulative damage have been proposed; one by Melby and Kobayashi in 1998 and the other by Hanzawa et al. in 1996. In this paper, comparison is made between the two methods for a Tetrapod armor layer. For the damage progression of Tetrapod armor units, hydraulic experiments are made. In the case where a severe damage occurs at the beginning of the life cycle of the breakwater, the two methods do not show significant difference, but in general the latter predicts a larger cumulative\ud
damage than the former. For a Tetrapod armor layer, it is recommended to use the average of the two methods

Kim, M.

Kim, S.W.

Suh, K.D.

Hasanuddin University Repository

 397 
 
 
Proceedings of the 7th International Conference on Asian and Pacific Coasts  
(APAC 2013) Bali, Indonesia, September 24-26, 2013 
 
 
 
CALCULATION OF CUMULATIVE DAMAGE OF TETRAPOD ARMOR LAYER 
 
K-D. Suh 
1
, M. Kim 
2
 and S-W. Kim 
3
 
 
 
ABSTRACT: In the performance-based design of a breakwater armor layer, it is often necessary to calculate its damage 
accumulated over the life cycle of the breakwater. Two methods for calculating the cumulative damage have been 
proposed; one by Melby and Kobayashi in 1998 and the other by Hanzawa et al. in 1996.  In this paper, comparison is 
made between the two methods for a Tetrapod armor layer. For the damage progression of Tetrapod armor units, 
hydraulic experiments are made. In the case where a severe damage occurs at the beginning of the life cycle of the 
breakwater, the two methods do not show significant difference, but in general the latter predicts a larger cumulative 
damage than the former. For a Tetrapod armor layer, it is recommended to use the average of the two methods. 
 
Keywords: Armor layer; Breakwaters; Cumulative damage; Tetrapod. 
 
 
                                                 
1 Department of Civil and Environmental  Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-744, 
REPUBLIC OF KOREA 
2 Global Environment System Research Division, National Institute of Meteorological Research, 61 Yeoeuidaebang-ro, 16-gil, 
Dongjak-gu, Seoul 156-720, REPUBLIC OF KOREA 
3 Coastal and Hydraulic Laboratory, US Army Engineer Research and Development Center, 3909 Halls Ferry Road, Vicksburg, MS 
39180-3199, USA 
INTRODUCTION 
The existing stability formulas for armor units 
calculate the degree of damage of the armor layer during 
the peak of a design storm. In the performance-based 
design of the breakwater armor layer, however, it is 
often necessary to calculate its damage accumulated over 
the life cycle of the breakwater. There are two methods 
appeared in literature to calculate the cumulative 
damage; Melby and Kobayashi (1998) and Hanzawa et 
al. (1996) (hereafter referred to as Method 1 and Method 
2, respectively).  
The Method 1 was proposed for stone armor layers 
for which the damage level S was used, the physical 
description of which is the number of cubic stones with a 
side of 50nD , eroded within the width (along the 
breakwater alignment) of one nominal size 50nD . On the 
other hand, the Method 2 was proposed for Tetrapod 
armor layers for which the damage is expressed in terms 
of relative damage 0N , which is defined as the number 
of displaced Tetrapods within the width of one nD . 
Since S and 0N have essentially the same physical 
description, the two methods could be used for both 
stone and concrete armor units. 
Melby and Kobayashi (1998) conducted hydraulic 
experiments for damage progression of quarry stones. 
However, the experiments for concrete armor units are 
scarce. Suh and Chang (2003) compared the two 
methods with the experimental results of Melby and 
Kobayashi (1998) for stone armor units, but they 
compared only numerical results for Tetrapod armor 
units. In the present study, we conduct hydraulic 
experiments for damage progression of Tetrapod armor 
units. The two calculation methods are then compared 
for Tetrapod armor units. 
 
CALCULATION METHODS 
 
Method 1 
Based on the stability formula of Van der Meer 
(1987), Melby and Kobayashi (1998) expressed the 
damage level as 
 
    bS wS A N                                                                  (1) 
 
where 
5
50( / ( ))S S s nA a H D  , sH = significant wave 
height, / 1a    with a = density of armor units 
and  = fluid density, wN = number of attacking waves, 
and Sa  and b  = empirical coefficients. With the 
duration of wave attack given by m wt T N , where mT  = 
mean wave period, Eq. (1) can be written as 
 
b
S
m
t
S A
T
 
  
 
                                                          (2) 
 
To calculate the cumulative damage level in real 
situations of sH and mT varying with time, Melby and 
Kobayashi (1998) proposed an empirical procedure, in 
Calculation of Cumulative Damage of Tetrapod Armor Layer 
 
398 
 
which the damage level at arbitrary time it , iS , was 
expressed as 
 
    1 1( )i i
b b b
i i S m i iS S A T t t

                                           (3) 
 
where 1iS  = known damage level at 1it t  , and imT = 
mean wave period during the period from 1it t  to it t . 
Based on the empirical formulas of Suh and Kang (2012), 
the counterpart of Eq. (3) for Tetrapods can be written as 
 
   
1
0.5 0.5 0.5
0 0 1( )i i i iN m i iN N A T t t

                                    (4) 
 
The coefficient NA  is estimated as 
 
   
2
0.4 3.25
if
9.2
s
m
n
N m mc
H
D
A

 
 
 
  
 
 
 
                  (5) 
 
 
2
0.45 0.4(cot ) 0.85
if
5.0
s
m
n
N m mc
H
D
A
 
 
   
  
 
 
 
  (6) 
 
where tan  = slope of the armor layer, 
0tan / /m sH L   with 
2
0 / (2 )mL gT  , and 
 
 
 
1.25
0.5 0.25
0
0.5 0.25 0.45
0
9.2 / 3.25
5.0 / 0.85 (cot )
w
mc
w
N N
N N


 
 
 
 
              (7) 
 
To calculate mc  using Eq. (7), 0iN  is necessary, which 
is a priori unknown. We calculated mc  using 10iN   and 
1 1
/
i iw i m
N t T
 
  in Eq. (7) and calculated 0iN  using Eqs. 
(4)-(6). This 0iN  and /i iw i mN t T  were then used in Eq. 
(7) to calculate mc , and 0iN  was re-calculated using 
Eqs. (4)-(6). Damage to the Tetrapod armor layer is 
assumed to occur under rough sea conditions of the 
significant wave height greater than a critical value, scH , 
which may be defined as the wave height corresponding 
to zero damage in a stability formula. 
 
Method 2 
Hanzawa et al. (1996) proposed a slightly different 
method. Supposing the wave height during the period 
from 1it t   to it t  is isH  and the cumulative damage 
level up to 1it t   is 1iS  , the number of waves which 
attacked the breakwater up to 1it t  , 'wN , is 
determined using 
is
H  and 1iS  , respectively, in places of 
sH  and S  in Eq. (1) as 
 
   
1/
1'
i
b
i
w
S
S
N
A

 
  
 
 
                                                         (8) 
 
The cumulative damage level up to it t , iS , is 
calculated by Eq. (1) with '
iw w w
N N N   and 
is s
H H  
where 
iw
N  = number of waves between 1it t   and 
it t . With 1( ) /i iw i i mN t t T  , iS  is given by 
 
   
1/
1 1
i
i i
b
b
i i i
i S
S m
S t t
S A
A T
 
    
    
   
                               (9) 
 
Using the similar procedure, the cumulative relative 
damage of Tetrapods is calculated by 
 
   1
0.5
2
0 1
0
i
i i
i i
i i
N
N m
N t t
N A
A T
 
    
    
   
                    (10) 
 
HYDRAULIC EXPERIMENT 
Hydraulic experiments for damage progression of 
Tetrapods were conducted in the wave flume at the 
Hydraulic and Coastal Engineering Laboratory in Seoul 
National University that was 36-m long, 1.0-m wide, and 
1.2-m deep. Fig. 1 shows the experimental setup. A 
horizontal bed with a 1/25 foreshore slope was installed 
at the elevation of 20 cm from the bottom of the flume. 
The breakwater model was placed at a distance of 25 m 
from the wave maker with the breakwater toe at a few 
centimeters from the beginning of the horizontal bed. 
The test section was divided into two channels by a 
vertical wall along the wave flume, each having a width 
of 0.6 m and 0.4 m, respectively. The breakwater was 
installed in the wider channel and the other channel was 
left empty. Irregular waves based on the modified 
Bretschneider-Mitsuyasu spectrum (Goda 2010) were 
generated with a piston-type wave maker. The water 
depth was 0.6 m at the wave paddle and 0.4 m at the toe 
of the structure. To measure the incident waves, three 
wave gauges were installed in the empty channel. The 
free surface displacements measured by these wave 
gauges were used to separate the incident and reflected 
waves using the method of Suh et al. (2001). Even 
though the channel is empty, wave reflection occurs 
from the sloping bed and the wave absorber located at 
the downstream end of the flume. The method of Suh et 
al. (2001) estimates the time series of surface elevation 
of the incident and reflected waves. The significant wave 
height sH  was calculated by the zero-crossing analysis 
of the time series of the incident wave profile.  
 
 
 K-D. Suh, et.al 
 
399 
 
 
 
 
Fig. 1 Sketch of wave flume and experimental setup. 
 
 
 
 
 
Fig. 2 Cross-section of breakwater 
 
 
 
 
Calculation of Cumulative Damage of Tetrapod Armor Layer 
 
400 
 
Fig. 2 shows the cross-section of the breakwater. 
The main characteristics of the Tetrapods were: height 
6.2TH   cm; nominal size 4.03nD   cm; 2.3a   
g/cm3; weight 150.5W   g; and layer thickness 8.0 cm. 
The underlayer consisted of stones of diameter 2.0-2.5 
cm and thickness of 6.0 cm, while the core consisted of 
stones of nominal size 50 1.3nD   cm. The slope of the 
structure was cot 1.5  . A little wave overtopping 
occurred when the significant wave height was greater 
than 18 cm. Tetrapods were placed in two layers: 
randomly placed upper layer on the regularly placed 
lower layer. Since the Tetrapods in contact with the 
sidewalls of the flume have less degree of interlocking, 
they were fixed not to move and were not included in the 
calculation of damage. 
Eight different irregular wave trains of 15 min 
duration with unique combinations of wave height and 
wave period were used to simulate storms of several 
hours’ duration. Table 1 summarizes the incident wave 
characteristics at the toe of the structure. The number of 
waves in the 15 min burst was approximately 450 to 710 
depending on the wave period. The surf similarity 
parameter based on mT  and sH , 
2 1/2tan [ / (2 )]
m m s
gT H   , was in the range 3.04-
4.40. On the other hand, the surf similarity parameter for 
collapsing waves, mc , calculated by Eq. (7) using 
various combinations of 0N  = 0.0, 0.5, 1.0 and wN  = 
1500, 3000, 4500 was in the range of 2.94-4.26. This 
proves that the waves listed in Table 1 produce intense 
collapsing breakers at the structure. 
 
Table 1 Test wave conditions. 
Wave sH  (cm) mT  (s) wN  m  
1 12.7 1.82 495 4.26 
2 11.8 1.27 710 3.08 
3 12.7 1.51 596 3.53 
4 12.7 1.43 630 3.34 
5 14.6 1.83 493 3.98 
6 13.6 1.46 617 3.29 
7 15.5 2.00 450 4.23 
8 15.6 1.61 560 3.39 
 
Five test series were conducted as shown in Table 2. 
Each series consists of the eight wave cases listed in 
Table 1 in different sequences, lasting 2 h with 
approximately 4550 waves. The test series were intended 
to compare cumulative damage caused by different 
sequences of storms as listed in Table 2. In series A, the 
storm intensity increased monotonically from wave 1 to 
wave 8. In series B and C, strong storms occur in the 
middle of the sequence, while in series D and E, strong 
storms occur at the beginning and end of the sequence. 
For each test series listed in Table 2, tests were repeated 
nine times. The tests of the maximum and minimum 
final damage were discarded, and the remaining seven 
tests were used to calculate the mean and standard 
deviation of damage. 
 
Table 2 Wave sequences in test series. 
Ser. A Ser. B Ser. C Ser. D Ser. E 
1 3 2 6 8 
2 5 7 5 6 
3 8 8 4 1 
4 4 4 3 4 
5 2 6 2 5 
6 7 1 1 2 
7 6 3 7 3 
8 1 5 8 7 
 
 
COMPARISON WITH EXPERIMENTAL 
RESULTS 
The calculation methods are applied to simulate the 
cumulative damage of the Tetrapod armor layer in the 
present experiment. Fig. 3 shows the relative damage 
calculated by Eqs. (4) and (10) along with the 
experimental data for series A to E. The experimental 
data are represented by error bars indicating 1  standard 
deviation. The data of zero standard deviation indicate 
no damage progression from the previous damage. The 
time interval of calculation was 15 min, which is the 
same as the duration of each wave case in the experiment.  
In general, the Method 2 predicts larger damages 
than the Method 1 except series E in which the first 
wave is the largest so that severe damage occurred at the 
beginning of the simulation. This trend is the same as 
that for stone armor layer (See Fig. 5 in Suh and Chang 
2003). In series A where the storm intensity increases 
monotonically, both methods initially over-predict the 
cumulative damages, but the agreement between 
prediction and measurement becomes better in the last 
stage where strong storms occur. In series B and C 
where strong storms occur in the middle of the sequence, 
the measured damage locates somewhere between the 
two predictions. In series D and E where strong storms 
occur at the beginning and end of the sequence, both 
methods well predict the cumulative damage, especially 
in series E where the first wave is the largest so that 
severe damage occurred at the beginning of the 
simulation. As a whole, it is difficult to judge which 
method better predicts the measurement. Since the 
measured damage locates somewhere between the two 
predictions except the case of large initial damage in 
which the two methods yield similar predictions, it is 
recommended to use the average of the two methods. 
 
 K-D. Suh, et.al 
 
401 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (a)       (b) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (c)        (d) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (e) 
 
 
Fig. 3 Comparison between calculated and measured relative damages of Tetrapod armors: (a) series A; (b) series B; (c) 
series C; (d) series D; (e) series E. 
 
 
 
 
 
Calculation of Cumulative Damage of Tetrapod Armor Layer 
 
402 
 
In the stability formulas for armor units, the damage 
level S  or the relative damage 0N  is proportional to 
wN  so that the coefficient b  must be 0.5. These 
equations were developed to calculate the damage of 
armor units during a single storm of relatively short 
duration. For rock armors, Van der Meer (1987) and 
Melby and Kobayashi (1998) showed that 0.5b   for 
long-duration tests. Melby and Kobayashi (1998) 
obtained 0.25b   by calibrating Eq. (3) with their long-
duration experimental data. In the comparison shown in 
Fig. 3 where severe damages occurred three or four 
times in each series, 0.5b   was used. This implies that 
the stability formula for Tetrapods developed for a single 
storm can be used for calculation of cumulative damage 
by several storms. If the methods are used for more 
number of storms, a calibration for the coefficient b  
may be necessary. However, the probability that a rubble 
mound breakwater will be severely damaged more than 
several times during its lifetime must be very low. 
 
 
CONCLUSION 
Comparison was made between the two methods for 
calculating the cumulative damage of Tetrapod armor 
units; Method 1 (Melby and Kobayashi 1998) and 
Method 2 (Hanzawa et al. 1996). In the case where 
severe damage occurred at the beginning of the lifetime 
of the breakwater, the two methods did not show 
significant difference, but in general the Method 2 
predicted larger cumulative damage than the Method 1.  
The two methods were compared against the 
experimental data of the present study. Depending on the 
time of occurrence of large storms, each method over- or 
under-predicts the measurement so that it is difficult to 
judge which method makes a better prediction.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Since the measured damage locates somewhere between 
the two predictions except the case of large initial 
damage in which the two methods yield similar 
predictions, it is recommended to use the average of the 
two methods for a Tetrapod armor layer. 
 
ACKNOWLEDGEMENTS 
This study was supported by the Brain Korea 21 
Project and the Project for Development of Reliability-
Based Design Methods for Port and Harbor Structures 
sponsored by Korea Ministry of Land, Transport and 
Marine Affairs. 
 
REFERENCES 
Goda, Y. (2010). Random Seas and Design of Maritime 
Structures, 3rd ed., World Scientific. 
Hanzawa, M., Sato, H., Takahashi, S., Shimosako, K., 
Takayama, T. and Tanimoto, K. (1996). New stability 
formula for wave-dissipating concrete blocks 
covering horizontally composite breakwaters. Proc. 
25th Int. Conf. Coastal Eng., 1665-1678. 
Melby, J.A. and Kobayashi, N. (1998). Progression and 
variability of damage on rubble mound breakwaters. J. 
Waterway, Port, Coastal and Ocean Eng. 124, 286-
294. 
Suh, K.D., Park, W.S. and Park, B.S. (2001). Separation 
of incident and reflected waves in wave-current 
flumes. Coastal Eng. 43, 149-159. 
Suh, K.-D. and Chang, E.-S. (2003). Comparison of 
calculation methods of cumulative damage of 
breakwater armor layer. Ocean and Polar Res. 25(3S), 
417-421 (in Korean). 
Suh, K.-D. and Kang, J.S. (2012). Stability formula for 
Tetrapods. J. Waterway, Port, Coastal and Ocean Eng. 
138, 261-266. 
Van der Meer, J.W. (1987). Stability of breakwater 
armor layers-Design formulae. Coastal Eng. 11, 219-
239. 
 
 


Calculation of cumulative damage of tetrapod armor layer

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Calculation of cumulative damage of tetrapod armor layer

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