Civil Engineering Reference
In-Depth Information
were employed to structural static analysis; others were used to analyze structural dynamic problems, and it focuses on the
latter here. For instance, linear dynamic behavior of bolted joint was modelled and its parameters were identified using
experimental observations by H. Ahmadian [
5
]. A simple, reliable and feasible finite element model with three-
dimensional solid elements about blind hole bolted connection was proposed to obtain bolt pre-tensional effect in
reference [
6
]. The identification of bolted lap joints parameters in assembled structures was done by H. Ahmadian and
H. Jalali [
7
]. Nevertheless, few suitable models and simplified analysis methods can be used for some specific problem.
Therefore, this article aims to offer the reader a good understanding for investigating the dynamics of the beam-like
structure with bolted connections.
In the present article, the foundational theory on structural dynamics is briefly reviewed, along with the influence of
stiffness and mass distribution on the beam's modal parameters (Sect.
15.2
). Modal testing on the beam-like structure with
different bolted connections is conducted; and effects of connection location and bolt preload on its modal parameters are
also obtained (Sect.
15.3
). A simplified but reliable finite element model of the bolted connection is built, and the spring
stiffness of the finite element model is optimized based on the data of modal testing (Sect.
15.4
). The conclusions of the study
are finally presented (Sect.
15.5
).
15.2 Beam Flexure
The formal mathematical procedure for considering the behavior of an infinite number of connected points is by means of
differential equations in which the position coordinates are taken as independent variables [
8
]. Inasmuch as time is also an
independent variable in a dynamic response problem, the formulation of the equations of motion in this way leads to
partial differential equations. Here, attention will be limited to one dimensional structures such as beam-type systems
which may have variable mass, stiffness and damping properties along their elastic axes. The partial differential equations
of these systems involve only two independent variables: time and distance along the elastic axis of each component
member.
15.2.1 Partial Differential Equation of Motion
For a straight, nonuniform beam, the significant physical properties of this beam are assumed to be the flexural stiffness
EI(x)
and the mass per unit length
m(x)
, both of which may vary arbitrarily with position
x
along the span
L
. The transverse loading
p(x, t)
is assumed to vary arbitrarily with position and time, and the transverse-displacement response
v(x, t)
is also a function
of these variables. For arbitrary boundary conditions, the partial differential equation of motion for the elementary case of
beam flexure [
8
] can be obtained as below
2
2
v
2
v
@
Þ
@
ð
x
;
t
Þ
Þ
@
ð
x
;
t
Þ
EI
ð
x
þ
m
ð
x
¼
p
ð
x
;
t
Þ
@
x
2
@
x
2
@
t
2
Of course, the solution of this equation must satisfy the prescribed boundary conditions at
x
¼
0 and
x
¼
L
. The free-
vibration equation of motion for this system is
@
2
Þ
@
2
v
ð
x
;
t
Þ
Þ
@
2
v
ð
x
;
t
Þ
EI
ð
x
þ
m
ð
x
¼
0
@
x
2
@
x
2
@
t
2
It can be found that the flexural stiffness
EI(x)
and the mass per unit length
m(x)
determine the modal parameters of the
structure. The natural frequencies, along with their mode shapes, can be obtained through solving this free-vibration
equation of motion. However, the solutions of the equations of motion for most complex systems generally can be obtained
only by numerical means.
