Metaheuristic Optimization in Seismic Structural Design and Inspection Scheduling of Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 156

Optimum Design Results considered; the modulus of elasticity ( E s and E c ),

the yield and cylindrical strength stresses ( f y and
f c ), the hardening parameter b of the stress-strain
curve and the cross-sectional dimensions ( B and
H ). One random variable is considered for both
confined and unconfined concrete. Furthermore,
one random variable is considered for the ground
motion excitation and one for the incident angle.
In order to account for the randomness of the in-
cident angle, the ground motions are applied with
a random angle with respect to the structural axes
uniformly distributed in the range of 0 to 180 de-
grees. The characteristics of the random variables
are given in Table 1, i.e. probability density func-
tion (PDF), mean value, coefficient of variation
(CoV) and type. Therefore, the total number of
random variables considered is: 54 (4+2 groups
of structural elements times 9 random variables)
for the eight-story RC building (since 4 groups of
columns and 2 groups of beams are considered)
and 45 (3+2 groups of structural elements times 9
random variables) for the five-story RC building
(since 3 groups of columns and 2 groups of beams
are considered) plus one random variable for the
seismic record and one for the incident angle.

For both formulations the designs variables of
the optimization problems are defined through
the dimensions of the columns' and beams' cross-
section. The columns are chosen to be rectangular
and they are grouped into four categories (C1, C2,
C3 and C4) for the eight-story test example while
they are grouped into three categories (C1, C2
and C3) for the five-story test example, while the
beams for both test examples are grouped into two
categories (more details can be found in a study of
Lagaros et al. (2004)). The two dimensions of the
columns/beams along with the longitudinal, trans-
verse reinforcement and its spacing are the five
design variables that are assigned to each group
of the columns/beams. Therefore, the structural
elements (beams and columns) are separated into
14 groups, 12 groups for the columns and 2 for
the beams, resulting into 70 design variables for
the eight-story test example; while for the five-
story test example the structural elements (beams
and columns) are separated into 10 groups, 8 for
the columns and 2 for the beams, resulting in 50
design variables in total.

Table 1. Random variables (Ellingwood et al., 1980, Dolsek, 2009)

Random
variable

Distribution (PDF) Mean CoV Type

Earthquake Uniform - - aleatory

Incident angle* Uniform - - aleatory

mean f c Lognormal 20 MPa 4% epistemic

f c Lognormal mean f c 15% aleatory

E c Lognormal 2.9×10 7 kN/m² 15% aleatory

Material mean f y (steel) Lognormal 500 MPa 4% epistemic

f y (steel) Lognormal mean f y 5% aleatory

E s (steel) Lognormal 2.1×10 8 kN/m² 5% aleatory

b (steel) Lognormal 1% 5% aleatory

b Normal design value 5% aleatory

Design variables

h Normal design value 5% aleatory

* In the Range of 0 to 180 degrees.