Civil Engineering Reference
In-Depth Information
where K is the stiffness matrix from the finite element model. This method is a weighted version of the EFI method where the
FIM is assembled using a Cholesky decomposition of the stiffness matrix given by
C
T
C
K
(14.4)
¼
where C is a upper triangular matrix. In this case, the FIM is defined as
T
FIM
SE
¼
Ψ
Ψ
where
Ψ
¼
C
Φ
(14.5)
The procedure is the same as the EFI algorithm but Guyan reduction is employed to reduce the stiffness matrix at each
step. This method is usually called SEMRO. Similar to the methodology based on the strain energy, the kinetic energy (KE)
could be employed. This method is called KEMRO.
14.3 Numerical Results
The effects of uncertainty on the OSP methodologies are studied with a truss bridge structure shown in Fig.
14.1
. A simple
two-dimensional finite element model was used to simulate the dynamic behaviour of the truss bridge. The finite element
analysis was carried out with the finite element code, CALFEM [
14
]. The bridge is discretized into 31 classical bar finite
elements with 2-DOF/nodes which results in a total of 14 nodes and 28 DOFs. The first ten mode shapes of the 2D-truss
bridge structure are considered as the target modes and the numerical analysis is performed with these modes.
14.3.1 Monte Carlo Simulation (MCS)
Uncertainty analysis can be performed with several stochastic methods available in the literature [
8
]. Monte Carlo
simulation (MCS) is the most accurate and popular uncertainty analysis technique [
9
,
11
,
12
]. The main advantage of
MCS is that uncertainty analysis can be performed without any modification of the deterministic finite element analysis
program. The MCS requires a large number of simulations and results in a high computational cost. However, the
computational cost or time for the truss type structure considered in this study is very small and therefore MCS is used in
this study.
The uncertainty analysis of truss bridge with MCS is discussed in the following paragraphs.
1.
Selection of the parameters under uncertainty
. In the real world, the truss bridge structure has several parameters which
are uncertain in nature. The parameters related to geometry and material properties of the bars can be uncertain because of
the error tolerances and thermal treatments during their manufacture, or variations in the ambient temperature. In the
present study, material properties and the cross sectional dimensions of the bars are considered as random variables with
Gaussian distributions. The mean values and coefficients of variation (CoV) of these uncertain parameters are selected
from Ref. [
14
] and given in Table
14.1
.
2.
Generation of samples for the stochastic parameters
. A simple random sampling scheme using MATLAB is used to
generate the random samples of the uncertain parameters. The Young's modulus (E), mass density (
) and cross section
area (A) of the bars are considered as independent random variables and the random samples are generated. Monte Carlo
ρ
Fig. 14.1
Two dimensional truss bridge
