2.
Single Degree of Freedom
Systems (1-DOFs)
under Harmonic Base Excitation
&
Elastic Response Spectra
Achilleas PAPADIMITRIOU
Assistant Professor of N.T.U.A.
February 2016
2.
Single Degree of Freedom
Systems (1-DOFs)
under Harmonic Base Excitation
&
Elastic Response Spectra
Achilleas PAPADIMITRIOU
Assistant Professor of N.T.U.A.
Suggested Reading:
Class Notes  Chapter 2
Α. CHOPRA  Chapters 1, 2 & 3 (A, B)
1
2.1 Equation of dynamic equilibrium
MY  CX  KX  0
Y  X U
MX  CX  KX   MU
για αρμονική διέγερση:
U  U O e it  U   2U O e it
and, finally:
MX  CX  KX  FO e it
με FO   2 MU O
Reminder: [eiΩt = cosΩt + i sin Ωt] …..
2.2 Relative displacement Χ
X1+X2
X2
X
t/T
X = X1+X2
where
Χ1 is the «transient» component, and
Χ2 is the «steady state» component of displacement
which is of greater practical importance (why?)
2
The “transient”component of displacement (Χ1) is a
damped free vibration described by the equation:
X 1  e t ( A cos  D t  B sin  D t )
The picture can't be display ed.
where:     C / 2 KM , n 
K
M
and D  n 1   2  n
e.g.
A=1
B=0
NOTE: This component has frequency equal to the fundamental frequency
of vibration of the 1-DOF (NOT the excitation frequency)
2.2 Relative displacement Χ
X1+X2
X2
X
t/T
X = X1+X2
where
Χ1 is the «transient» component, and
Χ2 is the «steady state» component of displacement
which is of greater practical importance (why?)
3
The “steady state” component of
displacement (Χ2) is harmonic,
With frequency equal to the
excitation frequency (Ω) and phase
difference (φ) from the excitation:
X 2  ( X ST  A )e i ( t  )
2
where X ST 
A
FO Ω 2 ΜU o  Ω 

   Uo
K
ω2 Μ
ω
1
[ 1  (  /  )2 ] 2  4 2 (  /  )2
tan  
2 (  /  )
1  (  /  )2
Question for the Class:
Based on the previous definitions and the figures for the
Dynamic coefficient Α and the angle of phase difference φ, describe
The basic mechanisms and the physical meaning of the RELATIVE
steady state response (X2/UO)MAX in the following benchmark cases:
I. Area Ω/ω <<<1.0
II. Area Ω/ω = 1.0
III. Area Ω/ω >>>1.0
Examples from engineering practice (and/or from every day life)
will contribute to better understanding and are welcome.
4
2.3 Total Displacement Υ
(velocity & acceleration)
Y  U  X  U O e it  X O e i ( t  )
  ( U O  X O e i )e it  YO e it
YO  ( U O  X O cos  )  i( X O sin  )
After normalization against the base excitation,
X
Y YO

 1  O e i ,
U UO
UO
2
XO   
  A
UO   
and finally,
or
2
YO
Y
 
i
[from a .. strange

 1    e
U UO
coincidence that is worth
 
proving analytically]
Y
Y
 O A
U
UO
Question for the Class:
Based on the previous definitions and the figures for the
Dynamic coefficient Α and the angle of phase difference φ, describe
The basic mechanisms and the physical meaning of the TOTAL
1-DOF response (Y/UO)MAX in the following benchmark cases:
I. Area Ω/ω <<<1.0
II. Area Ω/ω = 1.0
III. Area Ω/ω >>>1.0
Examples from engineering practice (and/or from every day life)
will contribute to better understanding and are welcome.
5
For harmonic excitation, all previous non-dimensional expressions for
relative and total displacements are also valid for the corresponding
velocities and accelerations. Hence, we may define the following two
“dynamic factors”:
AX 
X X
X


U U U
AY 
Y Y Y
 
U U U
AY
In this case also it is worth checking the physical meaning of the above diagrams for
the characteristic cases where Ω/ω<<<1.0, Ω/ω=1 and Ω/ω>>>1.
2.4 The “Elastic Response Spectrum”
Harmonic Excitation
The diagram for the dynamic factor variation ΑΥ – (Ω/ω) may be interpreted in
two basic ways:
A. For a given structure, i.e. when ω=2π/Τ=constant, it describes the maximum
dynamic response for various excitations with frequency Ω=2π/Τexc.
B. For a given excitation, when Ω=constant, it describes the maximum dynamic
response of various structures in terms of their fundamental vibration
frequency ω=2π/Τ, i.e. it provides the SPECTRUM (possible range) of the
visco-elastic structural response.
The second interpretation is of particular practical interest for the seismic design
of structures as it provides the maximum seismic actions (displacement, velocity,
acceleration) which are applied to the mass of the structure during base excitation.
Hence, it was extended to non harmonic – seismic excitations, in the form of the
widely used “Elastic Response Spectra”, initially by Biot (1932) and later by
Housner.
6
Elastic response ”SPECTRUM” for harmonic excitation
1.6
1.2
5
Excitation with
(Ü)max=0.30g
and
3
2
1
0
0
1
Ω/ω
2
0.8
0.4
Ω=31.4 rad/s
or
Τexc=0.20sec
3
0
0
0.1
0.2
T (sec)
0.3
0.4
Elastic response ”SPECTRUM” for harmonic excitation
1.6
The other way round…
i.e.
if the spectrum of a
harmonic excitation is
given to us…..
1.2
Sa (g)
ΑYΥ
R
4
Sa (g)
6
0.8
(inertia) acceleration applied to the
mass of a structure with period T
PGA=(Ü)max=0.30g
0.4
0
Τexc=0.20sec
0
0.1
0.2
T (sec)
0.3
0.4
7
Elastic response ”SPECTRUM” for SEISMIC excitation
Period, Τ
excitation
Elastic response ”SPECTRUM” for SEISMIC excitation
S a (g)
2
Lefkas earthquake
(14/08/2003
1.6
Μ=6.4
LONG
TRANS
1.2
0.8
0.4
Max.
acceleration of
seismic motion
0
0
0.4
0.8
1.2
Tstr (s)
1.6
2
Range of predominant excitation periods
8
Alternative types of Elastic Response Spectra
El Centro earthquake, ξ=2%
Max. relative displacement ΧΟ.
[Max. base shear force FΟ=K XO]
T, sec
Max. PSEUDO-velocity
VΟ=ω ΧΟ=(2π/Τ)ΧΟ
[Max. stored elastic energy
ΕΟ= ½ (Κ ΧΟ2) = ½ (ΜVO2)]
T, sec
Max. PSEUDO-acceleration
ΑΟ=ω2 ΧΟ=(2π/Τ)2ΧΟ
[Max. base shear force
FΟ=K XO = ΜΑΟ]
Comparison:
Relative velocity
Pseudo-velocity Vo
vs.
Pseudo velocity
max. relative

velocity X
o
9
Comparison:
Pseudoacceleration Ao
vs.
max. absolute

acceleration Y
o
Combined THREE-LOGARITHMIC
presentation
ΧΟ
VΟ=ω ΧΟ
ΑΟ=ω2 ΧΟ
10
Combined THREE-LOGARITHMIC
presentation
HWK 2.1:
For harmonic ground excitation, with
displacement amplitude UΟ=0.15m and
period Τexc..=0.40sec, compute:
ΕΙ=2x104 kNm2
H=5m, ξ=5%
1) The maximum absolute acceleration,
velocity and displacement at the floor
level of the frame.
2) The maximum relative dispalcement of
the floor, as well as the maximum shear
force Τ and moment Μ developing at the
top of the two columns.
Assume that the floor is rigid with a total
weight of 500 kN, while both columns are
flexible without mass.
11
HWK 2.2:
For harmonic seismic excitation, with displacement amplitude
UΟ=0.015m and period Τδιεγ.=0.40sec, compute the following
diagrams:
1) The elastic response spectrum for the relative displacement, for
ξ=2%.
2) The elastic response spectra for the Pseudo-velocity and the max.
relative velocity, for ξ=2% (in the same diagram with quest. 1)
3) The elastic response spectra for the Pseudo-acceleration and the
max. absolute acceleration, for ξ=2% (in the same diagram)
4) The ratio of the spectra of question 2 and the ratio of the spectra of
question 3 (in the same figure)
Compare with the respective diagrams – spectra for El Centro
earthquake. Discuss the similarities and the differences.
HWK 2.3:
Repeat HWK 2.1 for Lefkada
Earthquake
(any one of the two components)
2
ΕΙ=2x104 kNm2
H=5m, ξ=5%
LONG
TRANS
S a (g)
1.6
1.2
0.8
0.4
0
0
0.4
0.8
1.2
Tstr (s)
1.6
2
12
HWK 2.4:
A harmonic seismic excitation, with displacement amplitude UΟ and
excitation frequency Ω, is applied at the base of a SDOF system with
fundamental period of vibration equal to ω = K / M .
Draw the variation with Ωt of the base displacement (U/Uo - Ωt), the
relative displacement of the SDOF (X/Uo - Ωt) and its total
displacement (Y/Uo – Ωt), for the following characteristic conditions:
1) Ω/ω = 0 (very rigid SDOF)
2) Ω/ω = 1 (resonance)
3) Ω/ω →∞ (very flexible SDOF)
For convenience, you may use a large value of the damping ratio
(e.g. ξ=20%).
13