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Insights and Innovations in Structural Engineering, Mechanics and Computation – Zingoni (Ed.)
© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02927-9
Vibration of structures with variable stiffness
E. Demirkan & N. Kadioglu
Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey
ABSTRACT: The general assumption is that stiffness matrix is constant in structures. But if the loads
increase, deformation-load curve becomes nonlinear. The aim of this study is to analyze the vibration of a
structure under these facts. The chosen sample problem is a steel single storey frame with a horizontal force.
At first, classical stiffness matrix is determined under static loads. Then the structure has been reduced to a
unique mass-spring system with increasing loads. It is accepted that material is ideal elasto-plastic. The change
of stiffness under increasing loads will be calculated by plastic analysis. The system is not like the beginning
after first plastic hinge occurs and degree of freedom of the system changes. If the load increases the decreas-
ing of stiffness matrix continues. It is accepted that stiffness matrix is constant until first plastic hinge occurs.
Also, it is accepted that the stiffness matrix is also constant between first and the second plastic hinges. By this
way, a curve is obtained between load and stiffness matrix and the stiffness-force diagram has been plotted.
In addition, the positions of plastic hinges have been controlled by SAP2000 computer program. Finally, the
forced vibration of the sample structure under an harmonic load has been investigated. It is clear that the
external load varies by time. Then the force-time diagram and the stiffness-time graphic have also been plotted
under increasing loads. The damping coefficient of the system must be calculated for every stiffness value by
choosing a initial value for system damping. The displacement-time curve has also been given. The operations
of increasing loads have been repeated for decreasing loads. Also the displacement-time curves for increasing
and decreasing loads have been combined and they have been shown on a single diagram.
tion is made using certain models. The main fac-
tors that determine results, the distribution of
the masses in the structure, the characteristics of
external forces and displacements, internal fric-
tion, cracking, the resistances against deformation
of the bars(elements) and damping mechanism.
(Yerlici and Lus, 2007).
Consider a system which can only rotate about
an axis or can move in just one direction. This
model is called a single degree of freedom system.
The dynamic behavior of a structure under exter-
nal influences that varies depending on the mass of
system, the stiffness and energy loss in the system.
To solve a system under dynamic loads the equa-
tion of motion for the system is necessary. One
equation is sufficient to determine the dynamic
behavior of a single degree of freedom system.
2 SAMPLE PROBLEM
The chosen system is a plane frame with three bars
which is shown in the Figure 1. A1 and A3 ends
are connected to outer medium by built in connec-
tions. The chosen material is steel with young mod-
ulus 2.1 ×107 N/cm2 and density 7850 kg/m3 and
yield stress is 275 N/mm2. The material is assumed
as ideal elasto-plastic. Degree of freedom is three.
1 INTRODUCTION
The aim of this study is to calculate the variable
stiffness matrix under increasing and decreasing
loads. As the first step the solution of any struc-
ture under static loads is solved by classical matrix-
displacement method assuming stiffness matrix is
constant. And the stress resultants and stresses are
calculated in the bars forming the structure. Then
the loads have been increased. As a result of this
increment a specific cross section of a certain bar
cannot carry more moment. Then a plastic hinge
occur at this point. The new system is different
from the original system. Again load is increased
and new system is solved till a second plastic hinge
occurs. Here only plane systems are considered and
the moment at a plastic hinge is bending moment.
Degree of freedom of the first system decreases one
degree after each plastic hinge occurs. Then a specific
point is selected and the deformation of this point is
calculated for every stiffness matrix. These procedure
will continue till the system is unstable. And the load-
deformation curve is plotted for this specific point and
system is reduced a mass-spring system. And using
deformation-load curve the variation of the stiffness
of this imaginary spring by load is obtained.
After these same structure is examined under
dynamic loads which vary by time. Dynamic solu-
100
Cross sections are square 6 × 6 cm for beam and
8 × 8 cm for columns.
There is only a horizontal P1 force acting at
point B2. Horizontal displacement UB2 at point B2
is determined by solving this system via matrix-
displacement method. Here only this displacement
is considered then the stiffness of the system is
defined as
kP
UB
11
21
=
()
(1)
Then the P load is increased and at one P2 value
first plastic hinge occurs. It is assumed that stiff-
ness is constant till this value of P2. First hinge is
at A3 point.
This system is also solved and new stiffness k2
becomes
kP
UB
22
22
=
()
(2)
Then this procedure continues till last plastic
two hinges occur at A2 and B2. At this time this
value of P is named as limit load. The variation of
k versus P is given in Figure 2.
Then the dynamic loading is considered for the
spring-mass system given by Figure 3
For this system equation of motion is
mx cx kx Pt
++ =()
(3)
Here k is the stiffness, m is the mass, c is the
damping coefficient of the system and the force is
assumed as a harmonic load as follows
Pt Pt PTt
F
() cos( )cos()=−=−Ω∅ π∅
2
(4)
Here Ω is the frequency, TF is period and ∅ is
the phase angle which does not make any effect on
reactions of the system. Then this term is ignored.
The natural vibration of the system is also ignored.
The chosen TF is 0.4 sec for this problem.
Two new variables defined
c
m2=
ξω
(5)
k
m=
ω
2
(6)
Here ξ is damping ratio and ω is natural
frequency.
Three kinds of solutions can be found for equa-
tion (3) these are
· ξ > 1 heavy damping case
· ξ = 1 critical damping case
· ξ < 1 weak damping case
Here weak damping is considered and the solu-
tion of equation (3) becomes
αωξξ
12 2
1
,=− −
()
∓i
(7)
a
b
=−
=−
ξω
ωξ
12
(8)
xe bPtebtdtC bt
bPtebt
at at
t
at
=+
+
−
−∫
1
1
1
0
() cossin
() sinddt Cbt
t
+
∫2
0
cos
(9)
It is clear that the external load varies by time
in the dynamic loading. Therefore, the time val-
ues have been evaluated for increasing load values
using the load values at which the plastic hinges
occur.
Figure 1. Sample problem.
Figure 2. The variation of stiffness versus force.
Figure 3. Spring-mass system.
Insights and Innovations in Structural Engineering, Mechanics and Computation – Zingoni (Ed.)
© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02927-9
101
Mass of the system is taken as
mNsm=115 23 2
../
(10)
Then the force-time diagram has been plotted
under increasing loads in Figure 4.
After that, the stiffness-time graphic has also
been given which shows time intervals of this
stiffness.
Natural frequency values of the system have
been calculated for every time interval.
k
mrad
11
2149 698=→=
ωω
./sec
(11)
k
mrad
22
2249 668=→=
ωω
./sec
(12)
k
mrad
33
2349 659=→=
ωω
./sec
(13)
The damping coefficient of the system must be
calculated for every stiffness value by choosing a
initial
ξ
1
value for system damping.
ξ
1002=.
(14)
c
mcNsm=→=2 229 068
11
ξω
../
(15)
c
m=→=200200121
22 2
ξω ξ
.
(16)
c
m=→=200200157
33 3
ξω ξ
.
(17)
According to this, the natural frequency-time
and damping ratio of system-time graphics have
been plotted.
It is assumed that the stiffness matrix remains
constant between two values of the load when
using the stiffness-force diagram. The system has
been solved in a time interval that corresponds to
these loads. The values, at the end of the previ-
ous interval have been used as initial conditions to
determine the integration constants of the follow-
ing interval. And the displacement-time curve has
been plotted in Figure 8.
Table 1. The variation of load versus time.
Load (N)Time(sec)
P133210 0.095
P233225 0.096
P333300 0.1
Figure 4. The variation of force versus time.
Figure 5. The variation of stiffness versus time.
Figure 6. The variation of natural frequency versus
time.
Figure 7. The variation of damping ratio versus time.
Figure 8. The variation of displacement versus time.
Insights and Innovations in Structural Engineering, Mechanics and Computation – Zingoni (Ed.)
© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02927-9
102
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Figure 9. The variation of load versus time.
Figure 10. The variation of displacement versus time.
Table 2. The variation of load versus time.
Load(N) Time(sec)
P133300 0.104
P233225 0.105
P333210 0.2
The operations of increasing loads have been
repeated for decreasing loads. The changes of
increased and decreased loads depending on time,
has been represented in a single curve.
During calculations, the stiffness values of every
time range that is calculated for increasing loads
and decreasing loads have been taken as the same.
The displacement-time curve, for decreasing loads
has been plotted under these conditions. The dis-
placement-time curves for increasing loads and
decreasing loads have been combined and they
have been shown on one diagram.
3 RESULTS
The variation of the stiffness matrix by increas-
ing loads has been calculated. And this variation
is used for dynamic loading. The selected problem
has been reduced a single mass-spring system. But
if a set of system deformations is considered, stiff-
ness transforms to a matrix. Same procedure can
be extended to all calculations. But a restriction is
necessary. Whole exterior loads are horizontal and
have the same time dependencies.
Insights and Innovations in Structural Engineering, Mechanics and Computation – Zingoni (Ed.)
© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02927-9
