Optimum Design of Structures for Earthquake Loading by a Cellular Evolutionary Algorithm and Neural Networks - Structural Seismic Design Optimization and Earthquake Engineering - page 315

predicts the nonlinear time history responses of
the structure ( R new ).
The outline of the HNNS in training and normal
modes is shown in Figure 2.

considered. The mass of 10000 kg is lumped at
nodes of 1 to 4 of the structure. The 72 structural
members are divided into 9 groups, as follows: (1)
A1-A4, (2) A5-A12, (3) A13-A16, (4) A17-A24,
(5) A25-A28, (6) A29-A36 (7) A37-A40, (8) A41-
A48 and (9) A49-A72. The cross-sectional areas
of the elements can be chosen from the standard
Pipe profile list given as: (2.54, 11.2, 12.3, 13.9,
15.2, 17.2, 18.9, 21.4, 25.7, 26.4, 32.1 and 33.1)
10 -4 m 2 . As the nonlinear structural behavior is
considered only the displacement constraints are
included. In this case, the displacement of top
node of the structure is limited to 2 cm. The con-
straints are checked at 750 grid points with time
step of 0.02 seconds. The computational time is
measured in terms of CPU time required by a PC
Pentium IV 3000 MHz.
To train and test the HNNS a training set of
161 samples are randomly selected and their
natural frequencies and nonlinear time history
responses are evaluated by the conventional FEM.
During this process it is revealed that among all
the selected structures, in case of 11 ones the
nonlinear dynamic analysis does not converge.
Therefore, in the data there are 150 stable struc-
tures ( N S 1 =150) and 11 instable ones ( N S 2 =11).
The time spent to FE analysis of 165 structures
is 320 min.
A GRNN is trained to predict the natural fre-
quencies of the structures during the optimization
process. The inputs and outputs of the GRNN are
design variables ( X i , i =1,2,…,161) and frequen-
cies ( F i , i =1,2,…,161) of the selected structures,
respectively. Due to symmetry of the structure, its
1st, 3rd and 5th natural frequencies are considered.
From the 161 selected structures, 107 and 54 ones
are randomly selected to train and test the network,
respectively. In this case, the size of the GRNN is
9-110-3. The results of testing the generalization
of the GRNN are given in Table 1. The time spent
to train and test the GRNN is 0.6 min.
In order to train the PNN, 107 samples includ-
ing 100 stable and 7 instable structures are con-
sidered. Also to test the PNN, 54 samples includ-

Fundamental Steps of the
Methodology

As explained, in the proposed methodology
MCGA is employed to achieve optimization task
while HNNS is employed to predict the required
structural responses. Fundamental steps of the
methodology are as follows:

Step1: Data generation: A number of structures are
randomly selected and their natural frequen-
cies and nonlinear time history responses for
earthquake loading are computed.
Step2: Structural identification: a PNN is trained
to detect stable and instable structures based
on their natural frequencies.
Step3: Frequency evaluation: a GRNN is trained
to predict the natural frequencies of the
structures during the optimization process.
Step4: Nonlinear responses evaluation: another
GRNN is trained to predict the nonlinear time
history responses of the structures during the
optimization process.
Step5: Design optimization: optimal design
process is achieved by MCGA incorporat-
ing HNNS.

Numerical Results

To show the computational advantages of the pro-
posed methodology a 72-bar truss subjected to 15
seconds of the El Centro (S-E 1940) earthquake
record is designed for optimal weight. The struc-
ture is shown in Figure 3. The earthquake record
is applied in x direction. Young's modulus, yield
stress and mass density are 2.1×10 10 kg/m 2 , 2.4×10 7
kg/m 2 and 7850 kg/m 3 , respectively. A simple
bilinear stress-strain relationship with kinematic
hardening by the modulus of 6.3×10 8 kg/m 2 is