Elastic Design Response Spectra Uses Envelop of a computed peak dynamic response parameter for ! Characterize ground motions and assess demands on various types of single degree of freedom elastic simple structures. systems having a range of periods, for a given ground motion ! Basis for computing design displacements and forces in SDOF and viscous damping ratio and MDOF systems expected to remain elastic.
ma(t) + 2 !"v(t) + Kd(t) = -ma g (t) !
SD= umax
!=2% !=5% !=7%
Basis for developing design forces and displacements in nonlinear systems (two approaches): "
Modified elastic spectrum to account for nonlinearity
"
Equivalent elastic SDOF system
Period, sec.
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Design Response Spectrum Topics ! Developing design spectra from site specific ground motion time histories ! Selection of damping values ! Plotting formats ! Analytic relations for developing Elastic Design Response Spectrum " Deterministic #
#
"
#
Period, sec. 2.5
Median Median + 1
2
1.5
Statistical “attenuation” relationships Simplified empirical relationships (e.g., Newmark-Hall methods)
1
0.5
Sa
Uniform Hazard Spectrum #
SA
0 0
Basic approach (From USGS hazard maps used in current codes) Current spectra formulations found in codes (how do they relate to theory?)
0. 5
1
1. 5
2
2. 5
5% in 50 yrs.
#
Period
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3
Smooth Design Response Spectrum from Ground Motion Records !
Response Spectrum for actual ground motions are quite irregular. " Don t use individual spectrum for design " They can be used for analysis to assess response to a particular earthquake.
!
! !
SA="2SD
"
"
"
!
median + 1$ median
Period, sec.
Use suites of ground motions representing:
!
A specific deterministic design earthquake (e.g., M = 7 at 10 km) Match a stipulated design response spectrum (e.g., match code spectrum) A range of earthquakes types corresponding to the deaggregatized seismic hazard at the site.
The design response spectrum is obtained statically from all records.” The resulting “median” spectrum will be relatively smooth. The COV COV or ! lnx Standard Deviation ( ! ) can be used to establish a design spectrum with a desired probability of exceedence. Note: Various programs do this automatically.
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Generate Smooth Spectrum from Records !
PEER NGA Database will search for particular types of records and plot scaled response spectrum. Can download tables of spectral values for different periods and damping ratios
!
Bispec and other programs Permit user to input a suite of ground motion records and will find median and median plus ! x x " " ! values values
SA="2SD
median + 1$ median
Period, sec.
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Viscous Damping Viscous damping is a convenient analytical concept to account for general energy dissipation in a system and analytical uncertainties. " Friction between and with structural and non-structural elements. Localized yielding due to stress concentrations and residual stresses under low loading and gross yielding under higher loads. " Energy radiation through foundation.
Damping is generally a function of: " " " "
"
"
Aeroelastic damping. " Viscous damping. " Analytical modeling errors. "
" "
Material Amplitude (stress) Type of nonstructural elements Type of foundations and supporting soils Frequency Type of connections Complexity of model (different parts of structure will be responding differently)
Constant viscous damping is a simplification. Damping can produce substantial forces that are only crudely modeled compared to inertial and restoring forces.
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Data on Viscous Damping 40 From: Hashimoto et al Data for Welded Steel Moment Frames, From Hashimoto et al, 1992
% , 30 o i t a R g n i p20 m a D s u o c s 10 i V
0
?
0
0.5
1.0
1.5
References ! NRC, "Regulatory Guide 1.61, Damping Values for Seismic Design of Nuclear Power Plants," U.S. Atomic Energy Commission., Oct. 1973. ! Coats,D., "Damping in Building structures During Earthquakes, Test Data and Modeling," NUREG/CR-3006, Jan. 1989. ! Hashimoto, P. et al, "Review of Regulatory Guide 1.61 Structure Damping Values for Elastic Seismic Analysis of Nuclear Power Plants," Nuclear Regulatory Commission, 1992
Stress Ratio, f/f y
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Recommended Design Damping Values !
!
Many codes stipulate 5% viscous damping, unless a more properly substantiated value can be used. Note that actual damping values for many systems, even at higher levels of excitation are less than 5%.
Structure Type
Welded Steel Bolted Steel Prestressed Concrete Reinforced Concrete
Working Stress Range (no more than about 1/2 yield stress) NRC 1.61
Coats
2 4 2
2 to 3 5 to 7 2 to 3
3.5 4.5 TBD
4
2 to 5*
4
At or Just Below Yield Point
Hashimoto NRC 1.61
Coats
Hashimoto
4 7 5
5 to 7 10 to 15 5 to 7
4** 6 TBD
7
7 to 10
7
* lightly cracked sections represent lower values in range ** friction bolted connections same as welded steel TBD: values to be determined when sufficient data is available
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Formats for Plotting Spectra ! Tripartite
A variety of formats used !
Log SV
S A-T, S V -T, S D -T and S E -T
SV
Period, sec.
SA
0.03 0.13
SV = "SD
T
Log SA
LogSD
Recall: Only SD vs T plotted here
SA = "SV = "2SD SD= umax
S A-SV -SD Format
Period, sec.
T=0.2 0.5 2.0
SA = "2SD
SE= Period, sec.
Log T
! S A-SD Format
SA
SE
T
Building Period, T 4.0 sec.
mSV2/2
6.0
Period, sec.
SD
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Basis of Tripartite Graph Paper S V = S A / " = S AT/ 2 # D = " S D = 2 #S T -1 "
"
Log SV
line of constant spectral acceleration has a slope of 1 on a log-log plot of S V vs. T line of constant spectral displacement has a slope of -1 on log-log plot of S V vs. T
100 in/sec
Log SA 10g
1g 0.1g 0.01g
Log T
Log SV
Log SV
Line of Constant Spectral Acceleration
100 in
Log SD
10 in
10 in/sec
Line of constant Spectral Velocity
1 in/sec
Line of Constant Spectral Displacement
1 in 0.1 in 0.01 in
0.1 in/sec
Log T 0.01sec 0.1sec
1sec
10sec
Log T
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Tripartite Response Spectrum
After Fig. A6.1, Chopra, Dynamics of Structures
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S A-S D Format !
!
! !
An alternative form of plotting SA spectra has been introduced recently and has started to appear in building codes. Intent is to plot information on acceleration (force) and displace- ment on same graph with out complexity of tripartite paper SA Based on: SA = "2SD " "2 = SA/ SD
Line of constant T2
SD T=0.2
Building Period, T
0.5 2.0
4.0 sec.
Used to interpret nonlinear response in conjunction with “Capacity Spectrum” and “Yield Point Spectrum” Methods -Discussed later
6.0
SD
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Analytic Relations for Developing Elastic Design Response Spectrum !
!
Deterministic Approaches " Statistical attenuation relations for a given magnitude, distance, soil condition, fault type, etc. " Simplified empirical methods by Newmark and others for a given peak ground acceleration Spectra based on Probabilistic Hazard Analysis " Uniform hazard methods (focus on USGS data) " NEHRP Tentative Provisions for Seismic Regulations for New Buildings
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Statistically Derived Design Spectra !
!
!
!
Bins of ground motions selected with similar soil conditions, fault mechanism, magnitude, distance, etc.) Response spectra generated and averaged. Regression analysis used to develop equations for median response spectrum and standard deviation Resulting equations can be used in a seismic hazard analysis to develop design response spectrum with a desired probability of exceedence.
For given M, soil, mechanism, r 2.5
Median Median + 1
2
1.5
1
0.5
0 0
0.5
1
1.5
2
2.5
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3
Many Investigators Western US " " " " "
References :
Abrahamson &Silva Boore, Joyner &Fumal Campbell Sadigh Spudich
!
Interactive Tool on OpenSHA
!
Seismological Research Notes, Vol. 68, No. 1, Jan.-Feb. 1997. Joyner and Boore, “Prediction of Earthquake Response Spectra,” USGS Open File Report 82-977, 1982.
!
Central and Eastern US (CEUS) " "
Adkinson & Boore Toro et al
!
Subduction Zones " " "
Anderson Atkinson & Boore Youngs
Crouse, “Ground Motion Attenuation Relationships for Earthquakes in the Cascadia Subduction Zone,” Earthquake Spectra, Vol. 7, No. 2, 1991.
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Typical Statistical Relations Joyner and Boore (1982) -- S V at 5%viscous damping.
Boore, Joyner and Fumal (1997) - S V at 5% damping
log S v (cm/sec) = + (M-6) + ( -6) 2 - log r
log S v (cm/sec) = b 1 + b 2 (M-6) + b 3 (M-6) 2 + b 4 r + b 5 logr + b 6 G B + b 7 G C
+ b r +cS !
Simple form, but imprecise definition of soil conditions and small number of ground motions considered.
!
Period extend to 4 seconds.
!
Damping = 5% only
where r = [d 2 + h 2 ] 1/2 and terms are defined on slide 6-14, and a table of period specific coefficients in cited reference. Note:
Larger or random component Periods # 2 seconds Damping = 5% only
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NGA Attenuation Relationships !
!
Same process described in slides 6-21 to 6-23 used for estimating peak ground acceleration at a site can be used to generate a smoothed response spectrum for a particular site (magnitude, fault type, soil type, distance, etc.) See class website for reports and spreadsheets.
Campbell and Bozorgnia, 2006
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Examples: Abrahamson & Silva 2.5
M=6.8, Soil, r = 3km
2
Median Median + 1
1.5
1
1.8
M=6.8, r = 3km, Median
1.6 0.5 1.4
Soil Rock
1.2
0 0
0.5
1
1.5
2
2.5
3 1
0.8
0.6
0.4
0.2
0 0
0.5
1
1.5
2
2.5
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3
Effect of Magnitude and Distance 1.4
r=1 km M = 6.8 3 km 10 km 20 km 40 km
1.2
1
0.8
0.6
0.4
1.4
M = 7.8 M = 6.8 M = 5.8
1.2
0.2
1
0 -0.2
0.8 0.3
0.8
1.3
1.8
2.3
2.8
3.3
0.6
0.4
0.2
Abrahamson & Silva, Soil, median values
r = 3 km
0 0
0.5
1
1.5
2
2.5
3
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Compare Various Attenuation Relations 1.4
Sadigh Abrahamson and Silva Campbell
1.2
1
Spudich Joyner & Boore
0.8
0.6
0.4
0.2
0 0
0.5
1
1.5
2
2.5
3
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Directivity Effects !
!
The fault normal component of motion generally is substantially worse than the fault parallel component. This is primarily true for T >1 sec. This depends on the direction of fault rupture relative to the site. If the fault ruptures toward a building site, the effect is worse.
Hypocenter
See: # Section 5.4.5.3 in Ch. 5 Bozorgnia and Bertero Text; Somerville papers on Class Reference List May result in need for increased design forces / displacements for long period structures close to faults (in one direction) #
Site Propagation
SANormal/SAave 2
0 45
1
90
SA
T, sec. Fault Normal Median Fault Parallel
T
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Directivity Effects (continued) The fault normal motion is increased and fault parallel motion is decreased compared to the average spectrum from an attenuation relation. " Broadband scaling Somerville, P. et al, Modification of empirical ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seismological Research Notes, 68, 199-222. "
Narrow Band Scaling Somerville, P., Magnitude scaling of the near fault rupture directivity pulse, Proceedings, Int. Workshop on Quantitative Prediction of Strong Motion ad the Physics of Earthquake Sources, Oct. 2001, Tsukuba, Japan
!
NGA Spudich, P. and Choi, B., Directivity in Preliminary NGA Residuals, Report on Lifelines Program Task 1M01, PEER, Nov. 2006.
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Simplified Empirical Relations to Construct Elastic Design Spectrum !
!
!
The complexity of the previous Basic Concept methods, and the limited SAmax number of records available &a= SAmax/PGA two decades ago, led many PGA investigators to develop simplified empirical methods for Period developing design spectrum SVmax from estimates of peak or effective ground motion &v= SVmax/vgmax parameters. Based on the concept that all Period spectra have a characteristic SDmax shape dg Many artifacts of this can be &d= SDmax/dgmax max seen in current code spectra Period
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Newmark and Hall Approach !
Need to know a g d g
max,
!
, v g
max
and
max
plus $ a , $ v and $ d
Get a g
max
Structural Response Amplification factors
from deterministic or
probabilistic site hazard analysis !
!
Get v g
and d g
max
max,
from:
"
site hazard analysis
"
empirical functions using a g
max
Estimating d g
is problematic,
max
but not generally important unless T is > 4 sec.
Damping
% 1 2 5 10 20
Median Structural Response Amplification Factors d
v
1.82 1.63 1.39 1.2 1.01
2.31 2.03 1.65 1.37 1.08
a
3.21 2.74 2.12 1.64 1.17
See: Newmark and Hall, “Earthquake Spectra and Design,” EERI Monograph, EERI, Oakland, 1982
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Newmark and Hall Elastic Spectra
Damping
%
Median Structural Response Amplification Factors d
1 2 5 10 20
1.82 1.63 1.39 1.20 1.01
v
2.31 2.03 1.65 1.37 1.08
a
3.21 2.74 2.12 1.64 1.17
Median plus one Response Aplification Factors d
2.73 2.42 2.01 1.69 1.38
v
3.38 2.92 2.30 1.84 1.37
a
4.38 3.66 2.71 1.99 1.26
See: Newmark and Hall, “Earthquake Spectra and Design,” EERI Monograph, EERI, Oakland, 1982
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Construction of N-H Spectrum #
Short period range(less than 0.03 sec): S A=a g max
!
Amplified acceleration range ( T equal and somewhat greater than 0.16 sec): Constant S A = $ a a g max
#
Note: ! S V = S A /
!
(constant S V proportional to 1/T on conventional S A versus T plot) S D = S A / 2 S D=4 2 S A /T 2 ] (constant S D proportional to 1/T 2 on conventional S A versus T plot)
Intermediate Period Range - Constant S V = $ v v g max
#
S A=2 S v /T]
Long Period Range - Constant S D = $ d d g
SA
max
#
Very long period Range - Constant S D = d g
SA=PGA
max
T
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Basic Tripartite Spectrum 100 in/sec
Log SV
100 in/sec
Log SV
SV = constant = SAT/2' SD =constant = SA(T/2')2
10g 10 in/sec
10 in
1g
10 in/sec
0.1g
1 in/sec
1 in/sec 1 in
0.1 in/sec
0.01 in
0.1 in/sec
0.1 in
Not to scale 0.01sec 0.1sec
1sec
SD SD(T
10sec
SA=PGA 0.01sec 0.1sec
Log T SD =constant
SA=const.
SA
1sec
SA=const.
SD=dg 10sec
Log T
SA(1/T SA(1/T2
(T2
SD
SA=PGA
SD=dg T
T
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Construct Elastic Newmark Spectrum !
Construct Ground Motion “Backbone” Curve using constant a g , v g & d g lines - Take lower bound on three curves (the solid line).
Response Amplification Factors ! Short Period (T # 0.03sec): S a =a g ! Transition ! Constant Amplified Acceleration Range (T $ 0.13 sec): S a = $ a a g ! Intermediate Periods: S v = $ v v g SA= &a ag
Log SV
Log SA
ag = constant
Log SV
LogSD
Log SA
ag = constant LogSD
SV=&v vg vg = constant
vg = constant
dg = constant
dg = constant 0.03 0.13
Log T
Log T
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Completion of Elastic N-H Spectrum !
!
!
Long Period Range: S D = $ d d g
Log SV
Very long period range: S D =d g (transition unclear)
Log SA
D
LogSD
C
B
E
Connect lower bound response lines.
Log SV
Log SA
A 0.03 0.13
ag = constant LogSD
SA
SV= v vg
B
Log T C
vg = constant
SD =
d dg
dg = constant 0.03 0.13
D E
A
Log T
T
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Example: N-H Elastic Spectrum Consider: a g = 0.5g & ) = 5% !
Using Newmark " s estimates, get ground skeleton curve: " "
!
v g = 24 in/sec d g =18 in
Damping
%
Get Amplified Structural Response Values (here for + 1! ) "
S a (for T $ 0.13sec ) = 2.71x0.5 = 1.36g
"
S v (for intermediate T) = 2.30 x 24 in/sec = 55.2 in/sec S d (for long T) = 2.01x18 in = 36.2 in.
"
1 2 5 10 20
Median Structural Response Amplification Factors
Median plus one Response Aplification Factors
d
v
a
d
v
1.82 1.63 1.39 1.20 1.01
2.31 2.03 1.65 1.37 1.08
3.21 2.74 2.12 1.64 1.17
2.73 2.42 2.01 1.69 1.38
3.38 2.92 2.30 1.84 1.37
a
4.38 3.66 2.71 1.99 1.26
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Example: N-H Elastic Spectrum Consider: a g = 0.5g & ) = 5% !
Using Newmark " s estimates, get ground skeleton curve: " "
!
Log Sv 55.2 in/sec
v g = 24 in/sec d g =18 in
"
"
D
LogSd
C
B
36.2 in.
E
1.36g
S a (for T $ 0.13sec ) = 2.71x0.5 = 1.36g S v (for intermediate T) = 2.30 x 24 in/sec = 55.2 in/sec S d (for long T) = 2.01x18 in = 36.2 in.
0.5g
A
Get Amplified Structural Response Values (here for + 1! ) "
Log Sa
0.03 0.13
Sa 1.36g
0.5g
Log T Sv
B
A
C
= Sa
max
Tc /2
max
55.2 in/sec =1.36gT/2 TC=0.66sec. Sv = 2 SdT-1 TD= 4.11 sec.
Sa(1/T
0.03 0.13
D
Sa(1/T2 T
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Aside Current IBC & NEHRP provisions very similar to Newmark s approach " Short period range straight line to:
Sa
!
T o = 0.2S a / S a 1
"
0.2
Intersection at “C” given by: T c =S a / S a 1
1
"
0.2
Sa
1.0
C
B
To=0.2Sa1/Sa D
Sa = Sa1/T To 0.2 Tc 1.0
Sa 1.36g
Sv
B
= Sa
max
S a varies with periods greater than 4 seconds (or tabulated value of T L )
0.5g
A
Sa= 4Sa1/T2
T /2 max c
C 55.2 in/sec =1.36gT/2 TC=0.66sec.
T c = S a /S a
0.2 1/T 2 for
0.2
T
0.2
1
0.2
TD= 4.0 sec.
0.5g
0.2
Note from simple algebra: S v = S a (1.0sec/2 # ) max 1 Substituting gives: S a (1.0sec/2 # ) = S a T c /2 # Or
Sa
Tc=Sa1/Sa
Sa(1/T
0.03 0.13
Sv = 2 SdT-1 TD= 4.11 sec.
D
Sa(1/T2 T
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Comments on N-H Spectra If you only need spectral values at a single period, the entire spectrum is not needed; you need only the least of the following three quantities (if T $ 0.13sec) " S A = $ a a g " S A = " S v g )/T V * = 2 # ( $v * 2 2 $ d " S A = " S D = (2 # /T) d g ! Note: Use the lowest S A obtained above using the period of the structure to compute S v (= TS A/2 # ) and S D ( = ( T/2 # ) 2 S A); do not use $ v v g and $ d d g for this! !
!
!
Reasonably straight - forward to construct a spectrum. Simple to see effects of design changes.
!
Newmark " s method basis of and consistent with good methods for developing nonlinear response spectra.
!
However, the data it is based on and the overall methodology is NOT as good as newer statistical/analytic methods
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Effect of Soil Conditions on Spectrum !
For soft soils, a g remains the
Log Sv
same or decreases relative
D
Log Sa
LogSd C
B
E
to firm soil, but v g and d g increase (as suggested by Mohraz, etc.). Soil Type V/A Newmark and Hall 48 Rock 24-27 <30 ft. alluvium over rock 30-39 30 - 200 ft alluvium 30-36 Alluvium 48-57
Firm
A 0.03 0.13
2
AD/V
Sa
Soft Log T
C
B
Alluvium
6
D
5.2-5.3 4.2-5.3 3.8-5.1
A
Rock Firm T
3.5-3.9
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Observations from N&H for SDOF System in the Constant SV Range Sv = 2 S D T -1
S V = S AT/2 !
V base = M S A= 2 MS V
/T
!
max
using T =2 [M / K] V base = S V
T/2
max
drops off in inverse proportion to period. !
= S D = S V
displacements increase linearly with period increase.
1/2
!
= S V [M / K] 1/2 max so decreases with decreasing mass or increasing stiffness.
[MK] 1/2
max
so V base decreases in proportion to square root of decreasing mass or stiffness.
using T =2 [M / K] 1/2
Change 0.5M 1.5M 0.5K 1.5K 0.5M 0.5K
0.71 1.22 1.41 0.82 1.0
V b a s e 0.71 1.22 0.71 1.22 0.5
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Uniform Hazard Spectrum !
!
Based on deaggretization of hazard at site, a spectrum can be constructed consistently representing the effect of all earthquakes expected over a period of time. USGS provides this data online.
2% in 50 yr. Uniform Hazard Spectrum
SA / g
SAS
SA = SA1/T SA1 PGA/g Period, sec.
0.2
To
Ts
T1
4
To = 0.2SA1/SAS TS = SA1/SAS Sa=4S A1/T2 for T > 4 sec
3g!
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New Code Response Spectra The IBC and NEHRP Recommended Provisions for Seismic Regulations for New Buildings have implemented this basic procedure for estimating site specific design response spectrum. "
"
"
"
It has been incorporated, with minor changes into the year 2000 International Building Code Based on a Max. Considered Earthquake (MCE) with a 2% probability in 50 year (2500 year recurrence interval). Detailed maps provide spectral ordinates at T of 0.2 and 1 sec. Being redone, using NGA relations
"
"
"
The Code Design Level is intended to be a 10% probability in 50 year event. However, the IBC (and NEHRP) code uses a single level indirect method (not PBE), so only one level of event is specified. Taken as 2/3 " s of the MCE event. For California, this relation is about correct, but for other areas results in too high of event. Lower standards for design are permitted in these areas (e.g., ordinary frames).
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Comments on NEHRP Spectrum !
Maps based on probabilistic estimates by USGS (for 2% in 50 years) "
!
Frankel et al, National seismic hazard maps. Documentation. USGS Open File Report 96-532, 1996 (updated in late 2002).
Modified for design purposes not to exceed "
"
"
"
!
!
!
D = design
Smaller of deterministic or probabilistic estimates 1.5 times median deterministic values for a characteristic event for a know fault 1994 UBC values (depends on version of NEHRP/FEMA documents)
Maps are for medium rock sites. Factors to account for soil conditions are included: Modified Design Spectral Values: 2/3 intended to reduce from 2/50 to SDS = 2/3 FaSs 10/50 hazard level SD1 = 2/3FvS1 where S s and S 1 are the spectral values for 5% damping at T = 0.2 and 1.0 sec. and Soil parameters: 0.8 < Fa < 2.5 0.8 < Fv < 3.5 Depending on type of soil
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Spring 2009
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NEHRP (FEMA 368) Soil Factors Soil Definitions
SA
C
B
Soft D
A
Firm
0.2
1.0
T
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NEHRP Spectrum !
!
Basic form looks like typical code, Newmark and Hall or uniform hazard spectrum. Corner points: To = 0.2SD1/SDS TS = SD1/SDS
!
SDS SD1 0.4SDS
V = C s W C s =S DS /(R / I) < S D1/(T R / I)
Value Depends on Code Used!
but Cs > 0.044SDs / I and for SDC E&F, Cs>0.5SD1/(R/I)
Minimum Force
Sa =SD1/T
Note: SD value are expressed as a fraction of g, not in/sec2 !
Use:
Spectral Response Acceleration / g
Minimum Force permitted for safety, Uncertainty related to P-) effects, and near-fault directivity effects
Period, sec.
0.2
To
Ts
T1
TL
Note: Sa= 4S D1 /T2 variation permitted for T > 4 sec In FEMA 450, TL varies with location
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Spring 2009
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Modification for other than 5% Viscous Damping !
!
!
Statistical methods and code spectra have only been generated thus far for 5% viscous damping. Newmark's factors can be used to modify statistically derived or other spectrum. Note that these factors are period dependent! Consider if we have a spectrum at 5% viscous damping and we would like it at x%.
Damping
Sa(T, x%) = Sa(T, 5%)/B(T,x%), so B(t,x%) = Sa(T, 5%)/Sa(T, x%) !
%
1 2 5 10 20
Median Structural Response Amplification Factors
Median plus one Response Aplification Factors
d
v
a
d
v
1.82 1.63 1.39 1.20 1.01
2.31 2.03 1.65 1.37 1.08
3.21 2.74 2.12 1.64 1.17
2.73 2.42 2.01 1.69 1.38
3.38 2.92 2.30 1.84 1.37
a
4.38 3.66 2.71 1.99 1.26
If the 5% damped S v value is 60 cm/sec on the descending branch, an estimate of the 2% Sv value is 60/(1.65/2.03) = 60/0.81= 78 cm/sec
CEE 227 - Earthquake Engineering U.C. Berkeley
Spring 2009
©UC Regents
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Modification for other than 5% Viscous Damping !
!
!
Statistical methods and code spectra Sa have only been generated thus far for x% Damping 5% viscous damping. Newmark's factors can be used to modify statistically derived or other spectrum. Note that these factors are 5% Damping period dependent! Period Consider if we have a spectrum at 5% B(T,x%) viscous damping and we would like it at x%. Sa(T, x%) = Sa(T, 5%)/B(T,x%), so
1
B(t,x%) = Sa(T, 5%)/Sa(T, x%) !
Period
If the 5% damped S v value is 60 cm/sec on the descending branch, an estimate of the 2% Sv value is 60/(1.65/2.03) = 60/0.81= 78 cm/sec
[&d(5%)/&d(x%)] [&v(5%)/&v(x%)] [&a(5%)/&a(x%)]
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Spring 2009
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FEMA 356 Damping Values Modify spectral values at 0.2 and 1.0 sec., and use the same method to construct curves " S As *= S AS /B s "
S A1*= S A1/B 1
(FEMA 356)
Newmark (constant acceleration range)
0.8
0.77
0.8
0.81
5
1.0
1.0
1.0
1.0
10
1.3
1.29
1.2
1.20
Effective Damping, % !
2
Bs
B1 (FEMA 356)
Newmark (constant velocity range)
NOTE: From previous slide, B1 based on Newmark s spectral values for different damping values, we would expect B1 for 2% damping to be 0.81 !
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