Civil Engineering Reference
In-Depth Information
genetic algorithm (GA) is adopted to determine the stiffness of spring elements. Considering the characteristics of above
finite element model, the bolted connection optimization model for the beam-like structure is set up as follows,
8
<
t
X
1
2
3
f
i
F
i
min
f
ð
k
x
;
k
y
Þ¼
c
i
i¼
1
s
:
t
:
k
x
0
;
k
y
0
:
0
:
95
F
i
f
i
1
:
05
F
i
f
1
f
2
f
3
The stiffness of spring elements
k
x
and
k
y
are used to simulate the friction and extrusion interaction between different
steel parts, respectively. Here,
k
x
and
k
y
are selected as the variables during the optimization process.
f
1
,
f
2
and
f
3
are the first
three natural frequencies of the beam-like structure form finite element analysis and each of them is a function of
k
x
and
k
y
.
F
1
,
F
2
and
F
3
are the first three natural frequencies of the beam-like structure obtained from modal testing.
c
i
is the weight
coefficient of different natural frequencies. The objective function is to minimize the difference of the first three natural
frequencies between finite element analysis and model testing. The constraint is that each natural frequency from finite
element analysis should converge to no more than a five percent (
5 %) difference with the corresponding natural frequency
from modal testing and
f
1
<
f
2
<
f
3
must be satisfied during the optimization process.
15.4.3 Optimization Results
In this section, the first three natural frequencies of the beam-like structure with 2.5 Nm bolt preload are used to optimize the
stiffness of spring elements at the joints, i.e.
F
1
¼
11.488 Hz,
F
2
¼
31.503 Hz,
F
3
¼
58.835 Hz. The weight coefficient is
selected as
c
1
¼
1 to balance the influence of different natural frequencies on the objective function. Genetic
algorithm is used to find the optimal value of the spring stiffness
k
x
and
k
y
for the structure with 2.5 Nm bolt preload. With
the optimal
k
x
¼
c
2
¼
c
3
¼
1.004 N/mm and
k
y
¼
334.919 N/mm, the first three natural frequencies of the beam-like structure obtained
from finite element analysis are
f
1
¼
58.768 Hz, which are very close to the data of
modal testing. Other natural frequencies can also be obtained based on the finite element model with the optimal spring
stiffness, for example,
f
4
¼
11.520 Hz,
f
2
¼
31.502 Hz, and
f
3
¼
116.170 Hz,
f
5
¼
165.550 Hz, and
f
6
¼
224.990 Hz, which also approach to those natural
frequencies from modal testing (F
4
¼
223.941 Hz).
A simple and reliable finite element model of the bolted connection is built through selecting proper stiffness of spring
elements at the joints, and this method can be used to obtain much more modal parameters of the beam-like structure than
modal testing. In a word, the reliable finite element model of the bolted connection can be obtained through model updating
and the modal parameters of the structure under arbitrary bolt preload can be analyzed by numerical means, which make the
design of engineering structures more efficient and economic.
114.544 Hz, F
5
¼
165.395 Hz, F
6
¼
15.5 Conclusions
The influence of the stiffness and mass distribution on the beam's modal parameters is analyzed through numerical means
firstly, that is, the larger local stiffness leads to the increment of the natural frequencies, and any changes at the nodal points
of specific mode shape have little influence on the corresponding frequency. Afterwards a series of modal testing is
conducted to estimate the modal parameters of the beam-like structure with four steel parts connected with each other
through bolts, and the results of the modal testing validate the conclusions from numerical analysis before. For the beam-like
structure with three bolted connections, the larger bolt preload increases the stiffness of connections, which changes the
structural modal parameters, but this influence of the bolt preload highly depends on the connection location. Finally, a
simple and reliable finite element model of bolted connection is built based on the data from modal testing, which makes the
design of structures with bolted connections much more efficient and economic. In a word, a good design of the bolted
connection is helpful to obtain the desirable dynamic characteristics of the whole structure.
