Corso progettazione strutture con smorzatori viscosi PHD DICAM UNIBO_lezione 2

1. Design of structures with added viscous
dampers: from theory to practice
PART 2:
An insight into the Seismic behavior of frame structures
equipped with passive dampers
Michele Palermo
Department DICAM –University of Bologna
Bologna 6-7-8 Giugno 2023

2. Outline
• Damping reduction factors
• Complex damping theory and
applications for adjacent buildings
connected by dampers
• Peak interstorey velocity profiles
• Non linear viscous damping ratio
• Behaviour factors for damped
structures
STEP 1 h=>x
STEP 2 cL=>x
STEP 4 cL=> cNL
STEP 1 htot=>x
STEP 3 vmax
Links with 5-step procedure
Today topics

3. STOCHASTIC BASED
DAMPING REDUCTION
FACTORS

5. Damping reduction factors
h x
Why so many available formulations ?
h x
10
5

h
x
 
  

  /
ground n
k T T

Palermo et al.
2016 SOILDYN
T-H based DRF
Proposed formula
Damped pseudoacc. spectra

6. Stochastic-based DRF
formulation

7. Stochastic-based DRF
formulation

8. Stochastic-based DRF
formulation
For given ξg and ωg g and introducing
parameters β ω/ωn and k =Tg /Tn
If a “white-noise” PSD (G0) is considered
instead ofthe K–T PSD

9. Code-like DRF formulation
ηK T for different values of k =Tg /Tn

10. Stochastic-based DRF
formulation

11. PEAK INTERSTOREY
VELOCITY PROFILES

13. Problem formulation
Lateral peak deformed
shape
The pseudo-velocity and pseudo-
acceleration profiles are here defined as
follows (for profile A):
Let’s assume a given peak lataral
deformed shape:
the base shear, Vbase, can be evaluated as
follows (profile A):
N-storey frame
z
i-th storey
m
k
A)
B)

14. Peak interstorey velocity profiles
b

15. Numerical analysis
Higher modes effects and correction factor M for drifts and velocities:
Peak displacement profiles Peak velocity profiles

16. Design formulas
Estimation of 1st storey peak interstorey
velocities:

17. GENERALIZED DAMPED
SDOF

18. Generalized SDOF

19. Generalized SDOF
A generic MDOF system with
added viscous dampers can be
schematized by a generalized
SDOF system, by appropriate
considerations of equilibrium of the
dynamic forces resulting from:
• Elastic forces
• Inertia forces
• Viscous dampers forces
An assumption of the peak
lateral deformed shape profile
d=displacement vector
z=lever arm vector
i=identity vector
f=mode shape

20. Generalized SDOF
From simple concepts of
mechanics, a system of forces
is in equilibrium if:
• The resultant force is null
• The resultant moment of
the forces is null
Global equilibrium:
G-SDOF systemS:
d=lateral deformed shape vector
Or in matrix form:
T)
R)
For example:
first mode shape
from modal
analysis
T)
R)
T)
R)

21. Generalized SDOF
Mechanical analogies:
Damping ratio of the two
genaralized SDOF:
T)
R)
Modal analysis:
If d=f

22. Generalized SDOF
Equivalent Damping ratios of
the rotational and translational
G-SDOF systems
General expression:
Uniform ST frame, linear peak lateral
displacement profile:
 
, 5 1
,
, 1
tot SPD
to
storey SPD s t
tep m N
N
c
c x 
   
 

From the 5-step procedure

23. Equal total size constraints
Relative errors assuming a linear along the height peak drift profile for
uniform ST frames

24. Equal total size constraints
Normalized first-mode
damping ratios for an MPD
and SPD system under equal
total size constraint:
• The relative efficiency of
the SPD vs MPD system
rapidly decreases as the
number of stories
increases

25. G-SDOF for other dampers
disposition
Inter-storey (IS) placement Fixed-point (FP) placement

26. G-SDOF for other dampers
disposition
Design formulas based on G-SDOF method

27. Limitations
Studied placements
Simulated snap-back dynamic tests

28. THEORY OF COMPLEX
DAMPING

30. Theory of complex damping

31. Theory of complex damping
Elastic forces
Inertia forces
Damping forces

32. Normal modes for proportional
damping
Natural modes are typically referred as ni

33. Complex modes: the state space
State equations and
Generalized eigen problem
State matrix

34. Complex frequencies

35. Complex modes

36. Natural frequency

37. Undamped natural frequencies
ni
ni

38. Complex vs normal frequencies

39. The imaginary damped SDOF

40. Energy transfer for the
imaginary SDOF
The response is similar to that of a MDOF
system, due to the effect of energy transfer

41. Energy dissipation
• Work done by Fi
• Work done by Fd
• Work done by Fs

42. Energy transfer vs dissipation

43. TH response of viscously
damped structure

44. Interpretation of complex modes

45. Interpretation of complex modes

46. Interpretation of complex modes

47. Light damping assumption

48. State space formulation: recap
d
  
Mu C u Ku 0
d
         
 
     
   
       
 
0 M u M 0 u 0
M C u 0 K u 0
 
  
 
u
z
u
it
i
u e

0

 
D I
1 1
d
 
 
 
  
 
 
M C M K
D
I 0

k k k
i
  b
  Re( )
k k k k
  x 
 
2
Im( ) 1
k k k k
b  x 
  
is the state vector
is the solution general form
is the dynamic matrix, I is the identity matrix.
The k-th complex frequency can be expressed as follows:
is called complex frequency and is, in general, a complex number.
Dynamic equation of motion
Dynamic equation of motion in the state space

49. ST frames with IS dampers
Telaio a 4 piani con smorzatore al piano 1 Telaio a 4 piani con smorzatori al piano
1 e 4
m
m
m
m
k
k
k
k
IS
c
m
m
m
m
k
k
k
k
IS
c
IS
c
c
tot

50. ST frames with IS dampers
Telaio a 4 piani con smorzatore al piano 1
Telaio a 4 piani con sistema MPD
c
tot

51. ST frames with IS dampers
ST frame with 1 damper
ST frame with 2 dampers
c
tot
m
m
m
m
k
k
k
k
IS
c
IS
c
m
m
m
m
k
k
k
k
IS
c

52. Adjacent buildings connected by
viscous dampers

53. Main system parameters
Fundamental Frequency ratio Mass ratio Normalized added dampers coefficient

54. Basic case: two-SDOF system

55. Complex frequencies
Damping ratios Undamped frequencies
Lightly
damped
Phisical
meaning

56. 2-SDOF structures

57. 2-MDOF structures

58. Proposed formula
Estimation of the minimum damping
reduction factor of the peak roof
displacement of the reference
building is proposed as a function of
the fundamental frequency ratio Ω:
The values of parameter c can be
calibrated from the results of the
numerical simulations.

59. NON LINEAR DAMPERS

60. Non linear visocus dampers
Equal dissipated energy in a sinusoidal
cycle of amplitude Umax
L
NL
 
 
, ,max max
, ,max max
4 ( )
1 / 2
( )
2 3/ 2 / 2
d L d
d NL d
in
E F U
E F U G
G





 


 

 
1 1
max
4
2
NL input
eq
n
c U G
m
 

x
 
 

By equating EdL=Ed,NL:
 
1
max
2
4
input
NL
L eq n
L
U
c
c m
c G


x 

 
P
v
F
vP
vmax
FP
Fmax
0
 = 1.0
 = 0.3
Christopoulos,
Filiatrault

61. Smorzamento non-lineare
0
0,2
0,4
0,6
0,8
1
1,2
0 1 2 3 4
damping
ratio
Umax/Umax,design
Equivalent damping ratio as a
function of Umax normalized for
the design displacement:
eq
x
1 1
max
4
2
NL input
eq
n
c U G
m
 

x
 
 

Design
point
SLV
Demand
SLD
Demand
SLC

62. Numerical Simulations
CNL CL
5% DAMPED
Data:
cNL=782 (kN e m) calibrated for:
• x,target=22% (Tinput=1 s e Tn=1 s)
• cL=2828 (kN e m)
Investigated damped SDOF

63. Numerical Simulations
Investigated damped 3-storey planar frames

64. Analitical formula vs NLTH
analysis
0
0,2
0,4
0,6
0,8
1
1,2
0 1 2 3 4
Equivalent
damping
ratio
Umax/Umax,design
3-DOF SAP2000
SDOF SAP2000
ANALITICO

65. BEHAVIOUR FACORS FOR
DAMPED STRUCTURES

67. Problem formulation
We search x so that: ≈
Problem: we want to evaluate the behavior factor of a damped structure
that guarantee the same level of structural safety (C/D) of the
corresponding undamped (e.g. x=5%) structure

68. Problem formulation
Condition of EQUAL
STRUCTURAL SAFETY
≈
In general:
If ≈
x1.0
x≠1.0
Non-linear EP damped SDOF system

69. Parametric analysis

70. Results
Correction factor:
Correction factors vs T for damped EP SDOF systems:

71. Code-like formula
Overall damping
reduction factor:

72. Applicative Example
The RC structures
(naked and damped)
are designed
assuming:
• A force reduction
factor R= 4,
• A Viscous Damping
ratio x=30%