6. (3)
Table. 1: Range and statistics of the various parameters
Parameters
Data range Data statistics
Minimum Maximum Mean COV
C 0.3500 1.0000 0.6690 0.3456
Wc 0.1970 0.4430 0.3074 0.1913
ys 0.4920 1.9830 1.1344 0.2773
Fa 4.5005 9.3326 6.1568 0.1990
0.0001 0.0005 0.0003 0.3304
A nonlinear regression method in the MATLAB environment for the same dataset in the
present study was also implemented to get the scour depth prediction equation. It leads to the
following equation for the estimation of scour depth at the bridge abutment embedded in the bed of
the clay-sand mixture:
7. (4)
where, !
, non-dimensional maximum equilibrium scour depth; #$
%' , non-dimensional
bed shear strength;
() , abutment Froude number; C, is the clay content and WC, is the water
11. /
327 (the precision) less or equal to 0.01, then the precision is equal to zero, and fi= fmax = Ct.M. In this
case, M = 100 is used and, therefore, fmax = 1000. The advantage of this kind of fitness function is
that the system can find the optimal solution for itself [24, 25].
The second major step consists in choosing the set of terminals T and the set of functions F to
create the chromosomes. In this, the terminal set consists obviously of the independent variable(s),
(see table 2) but the choice of the appropriate function set is not so obvious, but a good guess can
always be done in order to include all the necessary functions[24, 25].
The third major step is to choose the chromosomal architecture, i.e., the length of the head
and the number of genes. A single gene and two head lengths were used initially and then, the
number of genes and heads were increased by one at a time during each run until the most
appropriate fit was obtained. It was observed that more than 7 genes and a head length greater than
twelve did not significantly improve the performance of GEP model. Thus, the head length, h = 12,
and 7 genes per chromosome were employed for the GEP model in the present study.
The fourth major step is to choose the linking function. In this study, addition was used as a
linking function and the final step is to choose the set of genetic operators that cause variation and
their rates. A combination of all genetic operators (mutation, transposition and crossover) is used for
this purpose (Table 2).
The explicit formulation of the GEP for the scour depth prediction at the bridge abutments in
the cohesive sediments has been obtained as:
89 :: ;
= ?@A BC