Civil Engineering Reference
In-Depth Information
3.3.3 Numerical Parametric Modeling to Calibrate Derived
Equations
Numerical parametric studies using finite element analysis were conducted to pro-
vide data to calibrate the expressions derived in Eq. (3.13, 3.24) and test their ac-
curacy. Lateral deflections were obtained by performing FEM analyses, using
Staad Pro
, on 28,000 different frame and load combinations in order to calibrate
the derived expression. The modeled frames have a varying amount of stories,
bays, size and location of the columns and beams as shown in Table 3.1. Eight of
the ten parameters were varied while the modulus of elasticity, E = 3150 ksi, and
Poisson's ratio,
were held constant at values suitable for concrete.
Frames varying from one to five stories and from one to ten bays were analyzed.
Three combinations of column height and beam length, and twenty-five combina-
tions of the moments of inertia were analyzed as shown in Table 3.1. Two differ-
ent values of total applied earth load,
W
, were analyzed for each of the four
pressure distributions (Fig. 3.1) as shown in Table 3.2. Three different combina-
tions of column height and beam length, and twenty-five different combinations of
the moment of inertias of the beams and columns were analyzed. Finally, 14,000
additional load frame combinations were analyzed in order to verify the principal
of superposition for the derived expressions. Lateral deflections were determined
from the FEM analysis at every story level of every analyzed frame, resulting in
120,000 data sets.
ν = 0.17,
3.3.4 Calibration Factors for Derived Equations
Multivariable nonlinear regression analysis (MNRA) was performed to calibrate
Eq. (3.13) using the regression analysis software
DataFit
(Oakdale 2002). In
MNRA, best-fit parameters for a model were obtained by minimizing the differ-
ence between all 120,000 of lateral frame deformations calculated using Eq. (3.13)
(model), and those obtained using FEM (data). The measurement of agreement of
the model and data is called the merit function, and is arranged so that small val-
ues represent close agreement between the data and the model. In MNRA the de-
pendence of the unknown calibration functions (
u
,
) is non-linear with
respect to the model, and the process of the merit function minimization is an
iterative approach. DataFit uses the Levenberg-Marquardt algorithm to adjust the
parameters. Applying the MNRA to Eq. (3.24), finds a calibration factor
u
= 3.
The equivalent area,
A
o
, is thus rewritten as:
α
,
β
, and
λ
30
(3.25)
A
=
0
3
1
l
+
c
I
I
(
)
()
b
+
1
b
c
b
l
l
c
b
