Response of Elastic SDOF subjected to Earthquake forces

1. CE-412: Introduction to Structural Dynamics and Earthquake
Engineering
MODULE 6
1
University of Engineering & Technology,
Peshawar, Pakistan
RESPONSE OF LINEAR ELASTIC S.D.O.F SYSTEMS TO
EARTHQUAKE LOADING

2. 2
“If a civil engineer is to acquire fruitful experience in a brief span of time, expose him to
the concepts of earthquake engineering, no matter if he is later not to work in
earthquake country.”
Nathan M. Newmark and Emilo Rosenbleuth

3. 3
Earthquake Response of Linear System
In this lecture, we will study the earthquake response of linear
SDOF systems subjected to earthquake excitations.
By definition, linear systems are elastic systems.
They are also referred to as linearly elastic systems in these
lectures to emphasize both properties.
fs
fs
u
Linear elastic system
Non-linear elastic system
No energy is dissipated
by system
u
Elastic-perfectly plastic
system (Elasto plastic system)
Non-linear inelastic system
Area enclosed by the
curve = Energy
dissipated by system

4. 4
As studied in module 5, acceleration at the base of a systems can be
replaced by the effective force, which in case of earthquake can be
written as effective earthquake force (indicated by the subscript “eff”.
Since this force is proportional to the mass, thus, by increasing the
mass the structural designer increases the effective earthquake force
Effective Earthquake Force
Effective earthquake force: horizontal ground motion
Base moving with
(t)
u
m
)
t
(
p g
eff



(t)
ug



5. 5
Strong motion record
Typical strong motion record
1 2 3 4 5

6. 6
Strong ground motions recorded in various earthquakes
t
g
u


Figure : Ground motions recorded
during several earthquakes.

7. 7
7
N-S component of horizontal ground acceleration recoded at El Centro, California
during the Imperial Valley earthquake of 1940
Ground acceleration,
Ground velocity,
Ground displacement,
g
u


g
u

g
u
Accelerogram used in these lectures

8. Equation of motion for SDOF system subjected
to EQ excitations
(t)
u
u
m
k
u
m
c
u
(t)
u
m
ku
u
c
u
m g
g









 








  n
n
cr
ω
m
k
and
2mω
c
c
Since 

 

(t)
u
u
ω
u
2ζ
u g
2
n
n 



 



 


9. 9 9
9
(t)
u
u
ω
u
2 ζ
u g
2
n
n 



 


 
The time variation of ground displacement, from the given time
variation of ground acceleration, can be determined by using any
appropriate time stepping numerical method.
Closer the time interval, more accurate will be solution. Typically, the
time interval is chosen to be 1/100 to 1/50 of a second, requiring 1500 to
3000 ordinates to describe the round motion of above given El- Centro
,1940, ground acceleration record having a duration of 30 sec.
Solution to equation of motion for SDOF
system subjected to EQ excitation

10. 10
The Holiday Inn, a 7-story, reinforced concrete frame building, was
approximately 5 miles from the closest portion of the causative fault during
1971 San Fernando Earthquake.
The recorded building motions enabled an analysis to be made of the
stresses and strains in the structure during the earthquake.
Dynamic amplification (Transmissibility effect ?)
TR ≈ 350/225=1.56

11. 11
Dynamic amplification (Transmissibility effect ?)
TR ≈ 65/30
TR ≈ 25/18

12. 12
Response quantities
Response is the structural system
reaction to a demand coming from
ground acceleration record
Thus a response quantity may be structural displacement,
velocity, acceleration, internal shear, bending moment, axial force
etc.
Sometime, the total acceleration, , of the mass would be
needed if the structure is supporting sensitive equipment and
the motion imparted to the equipment is to be determined.
t
o
u



13. 13
Response quantities
Pounding damage, Hotel de carlo, Mexico city, 1985 earthquake
One of the important response quantity is total lateral displacement
at the top end of structural system, , required to provide enough
separation between adjacent buildings to prevent their pounding
against each other during an earthquake
t
o
u

14. 14
Structural disp., , due to ground acceleration,
0.319g
ugo 


g
,
ug


in
u,
g
u


2%
ζ
0.5sec,
Tn 

u
El Centro,1940, ground acceleration
Corresponding relative displacement
at the top end of the SDOF frame

15. 15
(t)
u
u
ω
u
2 ζ
u g
2
n
n




 


 
The above given equation indicates that
Thus any two systems having the same values of Tn and ζ will have
the same deformation response u(t) even though one system may
be more massive than the other or one may be stiffer than the other
)
,
f(
u n

T

Influence of Tn and ζ on Peak displacement, uo ,
in a liner elastic SDOF system

16. 16 16
Effect of Tn on Deformation
response history
0.319g
go
u 


g
,
ug


El Centro ground acceleration
Response of SDOF systems with different values of Tn to
El Centro ground acceleration
In general, peak value of
displacement at the top end of a
SDOF increases with the increase
in the time period of the system.

17. 17
Effect of Time Period

18. 18 18
18
Effect of ζ on Deformation
response history
In general, peak value of
displacement at the top end of a
SDOF increases with the decrease
in the damping ratio of the system
0.319g
ugo



g
,
ug


El Centro ground acceleration
Response of SDOF systems with different values of ζ to
El Centro ground acceleration

19. 19
Approximate Periods of Vibration (ASCE 7-05)
Where h is the height of building in ft.

20. 20
Thus, structural systems with Tn=0.5sec, 1 and 2 sec may be considered
as 5, 10 and 20 story height buildings, respectively.
A building with 3 story height can be considered as Multi DOF system
with at least 3 DOFs.
To keep the discussion simple at this stage, it will be a reasonable
assumption to state that (out of 3 natural time periods of the 3 story
building) we consider only fundamental natural time period (Tn=0.3 sec)
to determine the response quantities for the building.
Later on we will discuss how all 3 vibration modes (and the
corresponding natural time periods) are calculated and are taken into
account to find the total response of a building with DOF =3
Because the empirical period formula is based on measured response of
buildings, it should not be used to estimate the period for other types of
structure (bridges, dams, towers).
Approximate Periods of Vibration

21. 21
Response spectrum concept
A plot of the peak value of a response quantity as a function of the
natural vibration period Tn of the system, or a related parameter such
as circular frequency ωn or cyclic frequency fn, is called the response
spectrum for that quantity.
Response is the structural system reaction to a demand coming
from ground acceleration record (i.e. Accelerogram) and when the
peak response commodities such as structural system displacement
, velocity and acceleration are plotted against the
structural system natural time period (or frequencies) will be called
spectrum
 
o
u  
o
u
  
o
t
u



22. 22
Response spectrum concept
Peak values of response quantities and shape of response
spectrum depends on the accelerogram
Each such plot is for SDOF system having a fixed damping ratio
ζ, and several such plots for different values of ζ are included to
cover the range of damping values encountered in actual structures.
The deformation response spectrum is a plot of uo against Tn for
fixed ζ. A similar plot for is the relative velocity response
spectrum, and for is the total acceleration response spectrum.
o
u

t
o
u



23. 23
Deformation response spectrum
Figure on next slide shows the procedure to determine the
deformation response spectrum. The spectrum is developed for
El Centro ground motions, as shown in part (a) of the figure.
The time variation of deformation induced by this ground motion
in three SDF systems is presented in part (b) of the figure
The peak value of deformation D ≡ uo, determined for SDF
system with different Tn is determined and shown in part (c) of the
Figure

24. 24 24
24
(a) El-centro ground acceleration; (b) Deformation response of three SDF systems
with ζ=2% and Tn=0.5,1, and 2 sec; (c) Deformation response spectrum for ζ=2%
Construction of deformation
response spectrum

25. 25
Pseudo-velocity response spectrum
Consider a quantity V for an SDF system with natural frequency ωn
related to its peak deformation D ≡ uo due to earthquake ground
motion:
D
T
2π
V
D
ω
n
n


The quantity V has the unit of velocity and is called relative pseudo-
velocity or simply pseudo-velocity. The prefix pseudo is used
because V is not equal to the peak velocity , although it has the
correct units.
o
u


26. 26
Pseudo-velocity response spectrum
T
2π
D
V
n

D
V
2%


2%


Tn D V=D*2π/Tn
0.5 2.67 33.6
1.0 5.97 37.5
2.0 7.47 23.5

27. 27
Pseudo-acceleration response spectrum
D
T
2π
D
ω
A
2
n
2
n 









D
A
2%


2%


Tn D A=D*(2π/Tn)2
0.5 2.67 1.09g
1.0 5.97 0.61g
2.0 7.47 0.191g

28. 28
Please note the following comments regarding pseudo commodities:
1. uo is same as D by definition.
2. Whereas is not taken as V, which by definition = ωnD
3. Similarly, is not taken as A which by definition= ωn
2D
o
u

o
t
u


A caution about Pseudo responses

29. 29
Displacement Response Spectra for Different Damping values
Higher the damping, the lower the relative displacement.
At a period of 2 sec, for example, going from zero to 5%
damping reduces the displacement amplitude by a factor of two.
While higher damping produces further decreases in displacement,
there is a diminishing return.
The % reduction in
displacement by going
from 5 to 20% damping is
much less that that for 0 to
5% damping.
Deformation response spectra for 1940 El-centro earthquake for different values of ζ

30. 30
Damping has a similar effect on pseudo acceleration. Note, however,
that the pseudo acceleration at a (near) zero period is the same for all
damping values.
Pseudo Acceleration Response Spectra for Different Damping
Values
This value is always
equal to the peak ground
acceleration, 0.319g, for
the ground motion in
question. i.e. El-centro
1940 earthquake
Pseudo Acceleration response spectra for 1940 El-centro earthquake for
different values of ζ

31. 31
Pseudo acceleration (A) Vs peak acceleration at top
The term Pseudo shall not be conceived by its meaning (i.e. false as
defined in English dictionaries). In fact it shall be taken as “an essence
similar effect to their relevant commodities ”
It can be observed from below graph that pseudo acceleration , A , and
peak value of true acceleration, have almost same values for systems
with Tn≤ 10 sec and .
It is worth mentioning that for elastic system the ζ seldomly exceed 5%
as such taking A same as negligible effect
t
o
u


0.1
ζ 
 
t
o
u


Tn- sec
t
o
u



32. 32
Pseudo velocity (V) Vs peak system’s velocity 
o
u

As shown in below graph that for medium rise buildings
(0.2≤ Tn ≤ 1 sec) as long as . Similarly
(0.2≤ Tn ≤ 3 sec) for
o
u
V 

0.1


o
u
V 

o
u
V 

o
u
85
.
0
V 

o
u
85
.
0
V 

0.1



33. 33
Combined D-V-A spectrum
The deformation, pseudo-velocity and pseduo-acceleration
spectra are plotted for a wide and practical range of Tn and for a
particular value of ζ .
The above mentioned procedure is repeated for different values
of ζ.
The results for different values of ζ over a wide range of Tn are
combined in a single diagram, called combined D-V-A diagram, as
shown on next slide

34. 34 34
34
Use of D-V-A spectrum
Figure: Combined D-V-A response spectrum for El Centro ground motion; ζ = 2%.
Note the values of D,V and A determined for a SDOF system with Tn=2 sec
Refer to slides 29, 31 and 32 for D,V and A, respectively

35. 35 35
35
Combined D-V-A spectrum
Combined D-V-A response spectrum for El Centro ground motion; ζ = 0,5,10 and 20%
For a given earthquake, small variations in structural frequency
(period) can produce significantly different results (See V value
for Tn = 0.5 to3 sec for El-centro earthquake)

36. 36
Relation between peak Equivalent static force, fso , and
Pseudo acceleration, A
o
so
k.u
f 
.m
ω
k
since 2
n

)
.u
m.(ω
.m).u
ω
(
k.u
f
o
2
n
o
2
n
o
so




m.A
fso



37. 37
Peak Structural Response from the response spectrum
As already discussed on previous slide, peak value of the equivalent
static force fso can be determined as:
mA
kD
f so


The peak value of base shear, Vbo, from equilibrium of above given
diagram can be written as:
m.A
f
V so
bo



38. 38
.w
g
A
.A
g
w
V
or bo


Where w is the weight of the structure and g is the gravitational
acceleration. When written in this form, A/g may be interpreted as
the base shear coefficient or lateral force coefficient . It is used in
building codes to represent the coefficient by which the structural
weight is multiplied to obtain the base shear
Peak Structural Response from the response spectrum

39. 39
The frame for use in a building is to be located on sloping ground as
shown in figure. The cross sections of the two columns are 10 in. square.
Determine the base shears in the two columns at the instant of peak
response due to the El Centro ground motion. Assume the damping ratio
to be 5%. The beam is too stiffer than the columns and can be assumed
to be rigid. Total weight at floor level = 10k
E = 3*103 k/in2
Problem M6.1
Solution

40. 40
5% Damped Elastic Displacement Response
Spectrum for El Centro Ground Motion
uo is decreasing with increase in Tn
0.67″
Solution (contd….)
Computing the shear force at the
base of the short and long columns.
 
 
g
76
.
0
ft/sec
51
.
24
)
12
/
67
.
0
(
/0.3
2
A
D
/T
2
A
7
6
.
0
D
u
2
2
2
n
o












Comments:
Although both columns go through equal deformation, however, the stiffer column
carries a greater force than the flexible column. The lateral force is distributed to the
elements in proportion to their relative stiffnesses. Sometimes this basic principle , if
not recognized in building design, lead to unanticipated damage of the stiffer elements.

41. 41
Response Spectrum normalized with peak ground parameters
thereby giving the amplification magnitudes for D,V and A e.g., for a
system with Tn = 0.5 sec and ζ =0.05. The amplification factors for
D,V and A are αD ≈ 0.2 , αV ≈ 1.9and αA≈2.3, respectively
Velocity amplification
factor
Acceleration amplification
factor
Displacement
amplification factor
D= ugo for
Tn>15 sec
go
u
A 


u= -ug
u= 0
Very rigid systems
Very flexible systems

42. 42
Spectral regions in Response Spectrum
c
n
go
T
T
for
const.
u
A




Therefore region from Tn = 0
to Tc is defined Acceleration
sensitive region.
Same logic apply in
defining velocity sensitive and
displacement sensitive regions

43. 43
Design Spectrum
Response spectrum cannot be used for the design of new
structures, or the seismic safety evaluation of existing structures due
to the following reasons:
Response spectrum for a ground motion recorded during the past
is inappropriate for future design or evaluation.
The response spectrum is not smooth and jagged, specially for
lightly damped structures.
The response spectrum for different ground motions recorded in
the past at the same site are not only jagged but the peaks and valley
are not necessarily at the same periods. This can be seen from the
figure given on next slide where the response spectra for ground
motions recorded at the same site during past three earthquakes are
plotted

44. 44 44
44
Figure: Response spectra for the N-S component of ground motions recorded at the
imperial valley Irrigation district substation, El Centro, California, during earthquakes
of May 18,1949;Feb 9,1956;and April 8,1968; ζ = 2%.
Tn = 0.4 sec
1.75
3.0
4.8

45. 45
Design Spectrum
Due to the inappropriateness of response spectrum as stated on
previous slide, the majority of earthquake design spectra are
obtained by averaging a set of response spectra for ground motion
recorded at the site the past earthquakes.
If nothing have been recoded at the site, the design spectrum
should be based on ground motions recorded at other sites under
similar conditions such as magnitude of the earthquake, the
distance of the site from causative fault, the fault mechanism, the
geology of the travel path of seismic waves from the source to the
site, and the local soil conditions at the site.

46. 46
Design Spectrum
For practical applications, design spectra are presented as
smooth curves or straight lines.
Smoothing is carried , using statistical analysis, out to eliminate
the peaks and valleys in the response spectra that are not desirable
for design. For this purpose statistical analysis of response spectra is
carried out for the ensemble of ground motions.
Each ground motion, for statistical analysis is normalized (scaled
up or down) so that all ground motions have the same peak ground
acceleration, say ;other basis for normalization can be chosen.
go
u



47. 47
Construction of Design Spectrum
Researchers have developed
procedures to construct such design
spectra from ground motion
parameters. One such procedure is
illustrated in given figure.
The recommended period values
Ta = 1/33 sec, Tb = 1/8 sec, Te = 10
sec, and Tf = 33 sec, and the
amplification factors αA, αV , and αD
for the three spectral regions (given
table on next slide), were developed
by the statistical analysis of a larger
ensemble of ground motions
recorded on firm ground (rock, soft
rock, and competent sediments).

48. 48
Construction of Design Spectrum (Newmark-Hall method)
Newmark and Hall have developed procedures to construct
such design spectra from ground motion parameters. One such
procedure is illustrated in given figure.

49. 49
Amplification factors for construction of Design
Spectrum

50. 50
Construction of Design Spectrum (firm soil)
We will now develop the 84.1 percentile design spectrum
for ζ=5%
For convenience, a peak ground acceleration is
selected; the resulting spectrum can be scaled by η to obtain the
design spectrum corresponding to
The typical values of
, recommended for firm ground, are used. For , these
ratios give
g
1
ugo 


ηg
ugo 


6
u
u
*
u
and
in./sec/g
48
u
u
go
2
go
go
go
go








g
1
ugo 


in
36
ugo 
and
in/sec
48
ugo 


51. 51
Using the values on
previous slide and values
given in table 6.9.2 (slide
53) for 84.1 percentile
and ζ =5%, the Pseudo-
velocity design spectrum
can be dawn as shown in
F i g u r e 6 . 9 . 4
3
.
2
v 

71
.
2
A

 01
.
2
D


Construction of Design Spectrum (firm soil)

52. 52
Displacement and Pseudo-acceleration design spectra can be drawn
from pseudo-velocity design spectrum using the relations being
already discussed and reproduced here for the convenience:
The Pseudo-acceleration and displacement design spectra drawn by
using above given equation are drawn in Figures 6.9.5 and 6.9.6 on
next two slides
n
n
n
n
T
2 π
V.
V ω
A
V
2 π
T
ω
V
D




Construction of Design Spectrum (firm soil)

53. 53
Construction of Design Spectrum (firm soil)
Acceleration sensitive region
Velocity sensitive region
Displacement
sensitive region

54. 54
Construction of Design Spectrum (firm soil)

55. 55
Design Spectrum for various values of ζ
Pseudo- velocity design spectrum for ground motions with
; ζ = 1,2,5,10 and 20 %.
in.
36
u
and
in/sec,
48
u
,
1
u go
go
go 

 

 g

56. 56 56
56
Pseudo- acceleration design spectrum (84.1 th percentile) drawn on log scale for ground
motions with ; ζ = 1,2,5,10 and 20 %.
in.
36
u
and
in/sec,
48
u
,
1
u go
go
go 

 

 g
Design Spectrum for
various values of ζ

57. 57 57
57
57
57
Figure: Pseudo- acceleration design spectrum (84.1 th percentile) drawn on linear
scale for ground motions with ;
ζ = 1,2,5,10 and 20 %.
in.
36
u
and
in/sec,
48
u
,
1
u go
go
go 

 

 g
Design Spectrum for various values of ζ

58. 58
Envelope Design spectrum
For some sites a design spectrum is the envelope of two different
elastic design spectra as shown below
Site
Nearby fault producing
moderate EQ
Far away fault
producing large EQ
Site
Nearby fault producing
moderate EQ
Far away fault
producing large EQ

59. 59
(a) A full water tank is supported on an 80-ft-high cantilever tower. It is
idealized as an SDF system with weight w = 100 kips, lateral stiffness k =
4 kips/in., and damping ratio ζ = 5%. The tower supporting the tank is to
be designed for ground motion characterized by the design spectrum of
Fig. 6.9.5 scaled to 0.5g peak ground acceleration. Determine the design
values of lateral deformation and base shear.
(b) The deformation computed for the system in part (a) seemed excessive
to the structural designer, who decided to stiffen the tower by increasing
its size. Determine the design values of deformation and base shear for the
modified system if its lateral stiffness is 8 kips/in.; assume that the
damping ratio is still 5%. Comment on how stiffening the system has
affected the design requirements. What is the disadvantage of stiffening
the system?
Problem M6.2

60. 60
Design Spectra (Building code of Pakistan 2007)
T= natural time period, Ca and Cv= seismic zoning coefficients that depends on
soil type and Zoning factor, Z

61. 61
Soil profile types (Building code of Pakistan 2007)

62. 62
Seismic Zone factor (Building code of Pakistan 2007)

63. 63
Seismic coefficients (Building code of Pakistan 2007)

64. 64
Design Response Spectra (Building code of Pakistan 2007)

65. 65
Base shear determination by static force method
(Building code of Pakistan 2007)
The total design base shear in a given direction shall be
determined from the following formula:
The total design base shear need not exceed the following:
The total design base shear shall not be less than the following:
In addition, for Seismic Zone 4, the total base shear shall also
not be less than the following:
Where V is Base shear, T= Fundamental natural time period, of structure
I= occupancy factor R= response modification factor (a measure of
ductility of structure) and W= Total weight of structure

66. 66
Determine the design values of lateral deformation and base shear for the
data mentioned in problem M 6.2. The tank is required to be constructed in
Zone 2B (as per BCP 07) on soil type SE. Consider elastic design and take
Importance factor, I= 1.0
Problem M6.3

67. 67
Solve following exercise problems (Chopra’s book, above
editions)
1. Problem 6.10
2. Problem 6.15
Further problem for practice:
6.12 to 6.14, 6.16 and 6.17
Problems for practice