-
13th World Conference on Earthquake Engineering Vancouver, B.C.,
Canada
August 1-6, 2004 Paper No. 1476
A SUMMARY OF FEMA 440: IMPROVEMENT OF NONLINEAR STATIC
SEISMIC
ANALYSIS PROCEDURES
Craig D. COMARTIN1, Mark ASCHHEIM2, Andrew GUYADER3, Ronald
HAMBURGER4, Robert HANSON5, William HOLMES6, Wilfred IWAN7,
Michael MAHONEY8, Eduardo MIRANDA9, Jack MOEHLE10, Christopher
ROJAHN11, Jonathan STEWART12
SUMMARY
The Applied Technology Council (ATC), with primary funding
provided by the Federal Emergency Management Agency (FEMA) and
supplemental support from the Pacific Earthquake Engineering
Research Center (PEER), is in the final stages of a project (ATC
55) to evaluate and improve the application of inelastic analysis
procedures for use with performance-based engineering methods for
seismic design, evaluation, and rehabilitation of buildings. The
project will culminate with the publication of FEMA 440:
Improvement of Nonlinear Static Seismic Analysis Procedures. This
paper provides a preview of the key conclusions of the work. The
focus is on anticipated recommendations to improve inelastic
analysis procedures as currently documented in FEMA 356 [1] and ATC
40 [2]. General categories of improvements include: ♦ Displacement
modification procedures (Coefficient Method) ♦ Equivalent
linearization procedures (Capacity Spectrum Method) ♦
Multi-degree-of-freedom effects ♦ Soil-structure interaction
effects 1 CDComartin,Inc, [email protected] 2 Santa Clara
University, [email protected] 3 California Institute of Technology,
[email protected] 4 Simpson Gumpertz and Heger,
[email protected] 5 University of Michigan,
[email protected] 6 Rutherford&Chekene,
[email protected] 7 California Institute of Technology,
[email protected] 8 FEMA, [email protected] 9 Stanford
University, [email protected] 10 University of California,
Berkeley, [email protected] 11 Applied Technology Council,
[email protected] 12 University of California, Los Angeles,
[email protected]
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2
The publication FEMA 440: The Improvement of Nonlinear Static
Seismic Analysis Procedures will be the product of the project.
This document will provide a review and discussion of simplified
inelastic seismic analysis of new and existing buildings. It will
contain guidelines for applications of selected procedures
including their individual strengths, weaknesses and limitations.
The document will also contain illustrative examples and expert
commentary on key issues.
EVALUATION OF CURRENT PROCEDURES Current nonlinear static
procedures estimate the global displacement response of a building
or structure utilizing a single-degree-of freedom representation of
behavior based on a nonlinear force-displacement relationship
(pushover curve) based on the monotonic static response to a
lateral load vector. The efficacy of procedures to predict SDOF
responses of a nonlinear oscillator can be investigated by
comparing the estimates to actual results from multiple nonlinear
responses history analyses in a statistical format. Parameters
affecting the maximum displacement of a SDOF oscillator and the
variations assumed for the evaluation study are summarized as
follows [3,4] (see Figure 1): Predominant hysteretic behavior
Figure 1: Basic hysteretic models used in the evaluation of
current procedures.
• The elastic-perfectly plastic (EPP) model is used as a
reference model. This model has been widely used in previous
investigations. This is a reasonable hysteretic model for steel
beams which do not experience lateral or local buckling or
connection failure. It is also a good model of
Elastoplasti c Perfectly Pla sti c Model
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100.00
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Fo
rce
Elastoplastic Perfectly Plastic
EPP
Elastoplasti c Perfectly Pla sti c Model
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Fo
rce
Elastoplastic Perfectly Plastic
EPP
Modified Clough - St iffness Degrading M odel
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Fo
rce
Modified Clough
SD
Modified Clough - St iffness Degrading M odel
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Fo
rce
Modified Clough
SD
Strength and stiffness degrading model
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0
200
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600
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Fo
rce
Strength and Stiffness Degrading
SSD
Strength and stiffness degrading model
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0
200
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600
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Fo
rce
Strength and Stiffness Degrading
SSD
Elastoplas tic Perfectly Plasti c Model
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Fo
rce
Nonlinear Elastic
NE
Elastoplas tic Perfectly Plasti c Model
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Fo
rce
Nonlinear Elastic
NE
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3
the behavior of other highly ductile systems including buckling
restrained braced frames (BRBF) and eccentric braced frames
(EBF).
• The stiffness-degrading (SD) model is representative of
well-detailed and flexurally-controlled reinforced concrete
structures whose lateral stiffness decreases as the level of
lateral deformation increases. In this model the unloading
stiffness is always the same as the initial stiffness.
• The strength and stiffness-degrading (SSD) model is aimed at
approximately reproducing the hysteretic behavior of structures
whose lateral stiffness and lateral strength decreases when
subjected to cyclic reversals. This model does not represent
systems that loose strength in the same cycle as yielding or
experience P-delta effects. This distinction in the type of
strength degradation is discussed below.
• The nonlinear elastic (NE) model unloads on the same branch as
the loading curve and therefore exhibits no hysteretic energy
dissipation. This model approximately reproduces the behavior of
pure rocking structures.
Basic global strength (R) In this study the lateral strength is
normalized by the strength ratio R, which is defined as
y
a
F
SmR
= (Eqn. 1)
where m is the mass of the system, Sa is the acceleration
spectral ordinate corresponding to the initial period of the system
and Fy is the lateral yielding strength of the system (see Figure
2). The numerator in (1) represents the lateral strength required
to maintain the system elastic, which sometimes is also referred to
as the elastic strength demand. Nine levels of normalized lateral
strength were considered corresponding to R=1, 1.5, 2, 3, 4, 5, 6,
7 and 8.
Figure 2: Global strength parameter, R
Period (T) Single-degree-of-freedom (SDOF) systems with periods
of vibration between 0.05s and 3.0s were used in this
investigation. A total of 50 periods of vibration were considered
(40 between 0.05s and 2.0s equally spaced at 0.05s and 10 periods
between 2.0s and 3.0s equally spaced at 0.1s). The initial damping
ratio, ξ, was assumed to be equal to 5% for all systems. Ground
motion A total of 100 earthquake ground motions recorded on
different site conditions were used in this study. Ground motions
were divided into five groups with 20 accelerograms in each group.
The first group (NEHRP [5] site class B) consisted of earthquake
ground motions recorded on stations located on rock with average
shear wave velocities between 760 m/s (2,500 ft/s) and 1,525 m/s
(5,000 ft/s). The second group (NEHRP site class C) consisted of
records obtained on stations on very dense soil or soft rock
with
y
a
F
SmR
=
mSa
Sd
Fy
T0
Demand spectrum
Capacity curve y
a
F
SmR
=
mSa
Sd
Fy
T0
Demand spectrum
Capacity curve
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4
average shear wave velocities between 360 m/s (1,200 ft/s) and
760 m/s (2,500 ft/s). The third group (NEHRP site class D)
consisted of ground motions recorded on stations on stiff soil with
average shear wave velocities between 180 m/s (600 ft/s) and 360
m/s (1,200 ft/s). The fourth group corresponds to ground motions
recorded on very soft soil conditions with shear wave velocities
smaller than 180 m/s that can be classified as type E. Finally the
fifth group corresponds to 20 ground motions influenced by
forward-directivity effects. The result of the variation in these
basic parameters was a database of 180,000 nonlinear response
history analyses representing the maximum displacement response of
a SDOF oscillator subject to earthquake motions. The accuracy of
the approximate nonlinear static procedures was determined by the
comparing the predictions to actual response histories as a
benchmark. Detailed results of the evaluation will be published in
FEMA 440. Selected illustrative results are included in this paper
in the subsequent sections in conjunction with proposed
improvements to the two NSP procedures.
STRENGTH DEGRADATION
It is important to distinguish between two different types of
strength degradation. Consider the hysteretic response of two
oscillators shown in Figure 3. While both exhibit inelastic
strength degradation, note that the first (cyclic strength
degrading) loses strength only in the cycles subsequent to that in
which it yields. The slope of the post-elastic portion of the curve
is not negative. The post-elastic stiffness, α, in any cycle is
zero or positive. In contrast, the other oscillator (in-cycle
strength degrading) has a negative α in the cycle in which the
yielding occurs.
Strength and stiffness degrading model
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0
200
400
600
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For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
αα = 0Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400Displacement
For
ce
Cyclic strength degradation In-cyclic strength degradation
Strength loss occurs in subsequent cycles;not in the same cycle
as yield.
Strength loss occurs in same cycle as yield.
αα = 0
Figure 3: Two types of strength degradation
The distinction between these two types of behaviors is
important because the dynamic response of the two oscillators
subject to earthquake motions can be radically different. The
results of the ATC 55 evaluation study, as well as other previous
research, demonstrate that cyclic-strength-degrading oscillators
(SSD in Figure 1) often exhibit maximum inelastic displacements
that are about the same or even less than those that do not lose
strength. The in-cycle strength-degrading counterpart, in contrast,
can be prone to dynamic instability particularly when subject to
ground motions that include large velocity pulses often associated
with near field records. If one were to generate a pushover curve
for each oscillator in Figure 3
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5
using the second-cycle backbone procedure of FEMA 356 there
might be very little difference, if any, between the two.
Additional studies were conducted as a part of the ATC 55 project
to confirm and illustrate this difference using an in-cycle
strength-degrading oscillator with characteristics as illustrated
in the right side of Figure 3. These studies confirm the potential
for dynamic instability and imply that structures with significant
negative post-elastic stiffness must have a critical minimum
strength to avoid collapse. This is discussed further in subsequent
sections.
DISPLACEMENT MODIFICATION The FEMA 440 document will propose
several improvements to the basic displacement modification
procedure in FEMA 356 [1]. These relate to the coefficient method
equation for the target displacement, δt for estimating the maximum
inelastic global deformation demands on buildings for earthquake
ground motions
gT
SCCCC eat 2
2
3210 4πδ = (Eqn. 2)
where the coefficients are currently defined as follows: Co =
modification factor to relate spectral displacement of an
equivalent SDOF system to the roof
displacement of the building MDOF system. C1 = modification
factor to relate the expected maximum inelastic displacements to
displacements
calculated for linear elastic response. C2 = Modification factor
to represent the effect of pinched hysteretic shape, stiffness
degradation and
strength deterioration on the maximum displacement response. C3
= Modification factor to represent increased displacements due to
dynamic P-∆ effects. Based on the analyses of the current
procedures two alternatives for improvement of the factor C1 are
were initially considered:
ALTERNATIVE 1: )R(c)T/T(a
Cb
ge
111
11 −⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡−
⋅+= (Eqn. 3)
SOIL PROFILE
a b c Tg (s)
B 42 1.60 45 0.75 C 48 1.80 50 0.85 D 57 1.85 60 1.05
ALTERNATIVE 2: )R()T/T(a
Cb
ge
11
11 −⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
⋅+= (Eqn. 4)
SOIL PROFILE
a b Tg (s)
B 151 1.60 1.60 C 199 1.83 1.75 D 203 1.91 1.85
These are both compared to the current definition in Figure 4.
Figure 5 illustrates the substantial improvement in error reduction
with either alternative
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6
Figure 4: Comparison of current and potential C1
coefficients
A simplified version of these alternatives is also under
consideration for inclusion in the final recommendations as
follows:
SIMPLIFIED ALTERNATIVE: 1 21
190
RC
T
−= + (Eqn. 5)
Figure 5: Comparison of mean errors for C1 coefficients for site
class C
The current definitions C2 and C3 are not clearly independent of
one another. C2 is intended to represent changes in hysteretic
behavior due to pinching, stiffness degradation, and strength
degradation. However, strength and stiffness degradation due to P-∆
effects are supposedly addressed by C3 as well. FEMA 440 will
recommend that C2 be used to modify displacements for purely cyclic
strength losses. In this case, example results indicate that the
current specification over estimates the actual effect of cyclic
degradation (see Figure 6).
W ITHOUT CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/( ? i)ex]
R=6.0R=4.0
R=3.0
R=2.0
R=1.5
SITE CLASS CTs = 0.55 s
W ITH CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /( ? i )ex ]
R=6.0
R=4.0
R=3.0
R=1.5
R=1.5
SITE CLASS C
Ts = 0.55 s
ALTERNATIVE I
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/(? i)ex]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 0.85 s
ALTERNATIVE II
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /(? i)ex ]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 1.75 s
Current Proposed
W ITHOUT CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/( ? i)ex]
R=6.0R=4.0
R=3.0
R=2.0
R=1.5
SITE CLASS CTs = 0.55 s
W ITH CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /( ? i )ex ]
R=6.0
R=4.0
R=3.0
R=1.5
R=1.5
SITE CLASS C
Ts = 0.55 s
W ITHOUT CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/( ? i)ex]
R=6.0R=4.0
R=3.0
R=2.0
R=1.5
SITE CLASS CTs = 0.55 s
W ITHOUT CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/( ? i)ex]
R=6.0R=4.0
R=3.0
R=2.0
R=1.5
SITE CLASS CTs = 0.55 s
W ITH CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /( ? i )ex ]
R=6.0
R=4.0
R=3.0
R=1.5
R=1.5
SITE CLASS C
Ts = 0.55 s
W ITH CAPPING
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /( ? i )ex ]
R=6.0
R=4.0
R=3.0
R=1.5
R=1.5
SITE CLASS C
Ts = 0.55 s
ALTERNATIVE I
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/(? i)ex]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 0.85 s
ALTERNATIVE II
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /(? i)ex ]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 1.75 s
ALTERNATIVE I
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/(? i)ex]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 0.85 s
ALTERNATIVE I
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app/(? i)ex]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 0.85 s
ALTERNATIVE II
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /(? i)ex ]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 1.75 s
ALTERNATIVE II
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[( ? i)app /(? i)ex ]
R = 6.0
R = 4.0
R = 3.0
R = 2.0
R = 1.5
SITE CLASS C
T g = 1.75 s
Current Proposed
SOIL PROFILE: CR = 5
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.3 0.5 0.8 1.0
PERIOD [s ]
C1
A lternative 1
A lternative 2
FEMA-356
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7
Figure 6: Comparison of current FEMA 356 C2 coefficients to
actual example results for
cyclic-strength-degrading oscillators
The following specification is being investigated as an
improvement for FEMA 440: 2
2
1
800
11 ⎟
⎠
⎞⎜⎝
⎛ −+=T
RC (Eqn. 6)
The plot in Figure 7 shows that this equation does not fall
below a value of 1.0. This bias is likely to be recommended since
the studies do not include the effects of the duration of shaking
that may be important for structures subject to cyclic strength
degradation.
Figure 7: An improved coefficient C2 as a function of period and
initial strength for cyclic-
strength-degrading behavior
Figure 8 illustrates a comparison between the current C3
specified in FEMA 356 with actual results of response history
analyses using oscillators that exhibit in-cycle loss of strength
and/or P-delta effects result in in a negative post-elastic
stiffness, α. It is clear that negative post elastic stiffness can
result in dynamic instability and collapse depending on the
magnitude of α, as well as initial period and initial strength.
Consequently FEMA 440 will recommend elimination of the C3
coefficient and introduce a minimum initial strength (maximum R)
requirement for systems with in-cycle strength degradation and
0.1
0.3
0.4
0.6
0.7
0.9
1.0
1.2
1.3
1.5
1.6
1.8
1.9
2.1
2.2
2.4
2.5
2.7
2.8
3.0
1
1.6
2.2
2.8
3.4
4
4.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
C2
T
R
2
2
1
800
11 ⎟
⎠
⎞⎜⎝
⎛ −+=T
RC
R
T
C2
0.1
0.3
0.4
0.6
0.7
0.9
1.0
1.2
1.3
1.5
1.6
1.8
1.9
2.1
2.2
2.4
2.5
2.7
2.8
3.0
1
1.6
2.2
2.8
3.4
4
4.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
C2
T
R
2
2
1
800
11 ⎟
⎠
⎞⎜⎝
⎛ −+=T
RC
R
T
C2
SITE CLASS B
0.0
1.0
2.0
3.0
0.0 0.5 1.0 1.5
PERIOD [s]
C2
Collapse Prevention
Life Safety
Immediate Occupancy
FRAMING TYPE 1
SITE CLASSES B(mean of 20 ground motions)
0.2
0.6
1.0
1.4
1.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]
C2=C1,SD/C1,E PP
R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5
Current (FEMA 356) Actual
SITE CLASS B
0.0
1.0
2.0
3.0
0.0 0.5 1.0 1.5
PERIOD [s]
C2
Collapse Prevention
Life Safety
Immediate Occupancy
FRAMING TYPE 1
SITE CLASSES B(mean of 20 ground motions)
0.2
0.6
1.0
1.4
1.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0PERIOD [s]
C2=C1,SD/C1,E PP
R = 6.0R = 5.0R = 4.0R = 3.0R = 2.0R = 1.5
Current (FEMA 356) Actual
-
8
significant P-delta effects. This minimum strength requirement
would also apply to solutions from equivalent linearization
procedures.
Current (FEMA 356)
T = 1.0s
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5R
C 3
α = − 0.21α = − 0.06
Actual
6
T = 1.0s
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6R
∆ i/∆ e
α = − 0.21α = − 0.06
1984 Morgan Hill, California EarthquakeGilroy #3, Sewage
Treatment Plant, Comp. 0°
Current (FEMA 356)
T = 1.0s
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5R
C 3
α = − 0.21α = − 0.06
Actual
6
T = 1.0s
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6R
∆ i/∆ e
α = − 0.21α = − 0.06
1984 Morgan Hill, California EarthquakeGilroy #3, Sewage
Treatment Plant, Comp. 0°
Figure 8: Comparison of current FEMA 356 C3 coefficients to
actual example results for in-
cycle strength-degrading oscillators
EQUIVALENT LINEARIZATION The capacity spectrum method documented
in ATC 40 [2] is a form of equivalent linearization based on two
fundamental assumptions. The period of the equivalent linear system
is assumed to the secant period and the equivalent damping is
related to the area under the capacity curve associated with the
inelastic displacement demand. ATC 40 also limits damping for
systems that exhibit strength and stiffness degrading behavior. The
average errors associated with the procedures are illustrated in
Figure 9. For non-degrading structures the current method
underestimates displacements, but generally over estimates for
structures with degrading behavior.
Figure 9: Errors associated with current ATC 40 nonlinear static
procedure
The focus of the ATC 55 effort to improve equivalent
linearization has been to develop better procedures to estimate
equivalent period and equivalent damping. This is an extension of
previous work [6,7] in which both parameters are expressed as
functions of ductility. These relationships are based on an
optimization process whereby the error between the displacement
predicted using the equivalent linear oscillator and using
nonlinear response history analysis is minimized [8,9].
Conventionally, the measurement of error has been the mean of the
absolute difference between the displacements. Although this seems
logical, it might not lead to particularly good results from an
engineering standpoint. This is
SIT E C LASS C
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[(∆ i)app/(∆ i)e x]
R = 8.0R = 6.0R = 4.0R = 3.0R = 2.0R = 1.5
APPROXIMATE: ATC40 - TYPE CEXACT: STRENGTH AND STIFFNESS
DEGRADING
Type C structures (severe degradation)
SIT E C LASS C
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[(∆ i)app/(∆ i)e x]
R = 8.0R = 6.0R = 4.0R = 3.0R = 2.0R = 1.5
APPROXIMATE: ATC40 - TYPE CEXACT: STRENGTH AND STIFFNESS
DEGRADING
Type C structures (severe degradation)SITE CLASS C
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
PERIOD [s]
E[(∆ i)a pp/(∆ i)ex]
R = 8.0R = 6.0R = 4.0R = 3.0R = 2.0R = 1.5
APPROXIMATE: ATC40 - TYPE AEXACT: ELASTO PLASTIC
Type A structures (non-degrading)
-
9
illustrated in Figure 10. It is possible to select linear
parameters for which the mean error is zero as for the broad, flat
distribution. However, the narrower curve might represent
equivalent linear parameters that provide better results from an
engineering standpoint, since the chance of errors outside a –20%
to +10% range, for example, are much lower. This is owing to the
smaller standard deviation in spite of the –5% mean error.
Figure 10: Illustration of probability density function of
displacement error for a Gaussian
distribution This general strategy has been applied to a series
of elasto-plastic, stiffness degrading, and
strength-and-stiffness-degrading hysteretic models generate optimal
equivalent linear effective periods and damping for a range on
periods and ductilities as illustrated in Figure 11. Also, shown in
Figure 11 are the current CSM specifications in ATC 40 [2].
Figure 11: New optimal effective (equivalent) linear parameters
for elastoplastic system.T0=0.1-
2.0 Using the results for discrete values of ductility, a curve
fitting process has leads to empirical expressions relating
effective period, Teff , and effective damping, ξeff , to
ductility, µ. Generally these expressions are
-
10
dependent on hysteretic type. However, reasonably good results
can be obtained for all types of behavior with the following
simplified expressions:
For 4.0µ < : ( ) ( )2 34.85 1 1.08 1 5effβ µ µ= − − − + (Eqn.
7) ( ) ( )2 3/ 1 0.167 1 0.0310 1eff oT T µ µ− = − − − (Eqn. 8)
For 4.0 6.5µ≤ ≤ : ( )13.6 0.318 1 5effβ µ= + − + (Eqn. 9) ( )/ 1
0.283 0.129 1eff oT T µ− = + − (Eqn. 10)
For 6.5µ > :
2
20
0.64( 1) 119.01 5
0.64( 1)eff
eff
T
T
µβµ
⎛ ⎞⎡ ⎤− −= +⎜ ⎟⎢ ⎥−⎣ ⎦ ⎝ ⎠ (Eqn. 11)
( )
( )
0.51
/ 1 0.89 11 0.5 1 1eff o
T Tµ
µ
⎡ ⎤⎛ ⎞−⎢ ⎥− = −⎜ ⎟⎜ ⎟+ − −⎢ ⎥⎝ ⎠⎣ ⎦
(Eqn. 12)
The solution to the two equations for effective period and
damping are illustrated in acceleration-displacement response
spectrum (ADRS) format in Figure 12. Note that the maximum
acceleration does not fall on the ADRS demand curve for the optimal
effective damping. As a part of the proposed improvements a
numerical transformation will be included to generate a modified
ADRS (MADRS) that will represent correct values on the acceleration
axis. This transformation facilitates the development of several
solution procedures that are very similar to those for the capacity
spectrum method in ATC 40. It should be noted that the
recommendations will include a limit on minimum strength as
discussed in the previous section.
Figure 12: Description of the Modified ADRS (MADRS) and its use
(from Iwan 2002).
-
11
MULTI-DEGREE-OF-FREEDOM EFFECTS In order to compare and
illustrate techniques for improving the results of nonlinear static
procedures related to the effects of higher modes, five example
buildings have been analyzed. The objective has been to compare
estimates made using simplified inelastic procedures with results
obtained by nonlinear response history analysis. The basic outline
of this effort was as follows: EXAMPLE BUILDINGS 3-Story Steel
Frame (SAC LA Pre-Northridge
M1 Model) 3-Story Weak Story Frame (lowest story at 50%
of strength) 8-Story Shear Wall (Escondido Village) 9-Story
Steel Frame (SAC LA Pre-Northridge
M1 Model) 9-Story Weak Story Frame (lowest story at 50%
of strength) GROUND MOTIONS 11 Site Class C Motions, 8-20 km, 5
events 4 Near Field Motions: GLOBAL DRIFT LEVELS Ordinary Motions
(scaled to result in specified
global drift) 0.5, 2, 4% for frames 0.2, 1, 2% for wall
Near-Field (unscaled) 1.8 to 5.0% for 3-story frames 1.7-2.1%
for 9-story frames 0.6 – 2.1% for wall
LOAD VECTORS/METHODS ILLUSTATED First Mode Inverted Triangular
Rectangular (Uniform) Code Adaptive SRSS Multimode Pushover
(MPA)
RESPONSE QUANTITIES (Peak values generally occur at
different
instants in time) Floor and roof displacements Interstory Drifts
Story Shears Overturning Moment
ERRORS Mean over all floors Maximum over all floors
The results of the illustrative examples are consistent with
previously published observations by researchers. It is apparent
that the approximate procedures can generally predict maximum
displacements reasonably well. Multi-mode pushover can also provide
good estimates of maximum inter-story drifts for some cases. But
beyond that, the nonlinear static procedures cannot provide
reliable estimates of MDOF effects. FEMA 440 will recommend
nonlinear response history analysis to determine these effects. As
a part of the MDOF an interesting and potentially promising
observation for future development has been made [10]. To generate
the example results, ground motions were scaled to give
pre-determined constant roof displacements for each case. This
effectively normalized the MDOF effects to the roof displacement.
In general it was noted that any single response history analysis
provided a better estimate of MDOF effects than any of the
approximate methods (see Figure 13). This suggests that seismic
hazard could be characterized by the maximum inelastic displacement
at the roof level. This could be determined for a structure with
nonlinear static procedures using the NEHRP [5] maps, for example.
When necessary, nonlinear response history analysis could be used
to investigate MDOF effects by using a small number of records
scaled to give the same roof displacement. This procedure could
avoid the both the necessity of generating a series of
spectrum-compatible records and the difficulty of combining results
of the analyses for practical use.
-
12
Weak—2 % Weak—4 %
0 50000 100000 150000 200000
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Floor
Overturning Moment (kips-ft)
2% Drift
0 50000 100000 150000 200000
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Floor
Overturning Moment (kips-ft)
2% Drift
0 50000 100000 150000 200000
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Floor
Overturning Moment (kips-ft)
4% Drift
0 50000 100000 150000 200000
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
Floor
Overturning Moment (kips-ft)
4% Drift
MedianCode SRSS
AdaptiveMultimode
Rectangular
Inverted Triangular
First ModeMin Max
Mean
SD SD
Overturning Moments— Weak-story 9-story frame
Figure 13:Selected results from the MDOF illustrative examples
(from Aschheim 2002)
SOIL-STRUCTURE INTERACTION EFFECTS
FEMA 356 [1] currently contains limitations (caps) on the
maximum value of the coefficient C1, the ratio of the maximum
inelastic displacement of a single degree of freedom elasto-plastic
oscillator to the maximum response of the fully elastic oscillator.
FEMA 356 includes the capping limitations for two related reasons.
First, there is a belief in the practicing engineering community
that short stiff buildings simply do not respond to seismic shaking
as adversely as might be predicted analytically. Secondly, it was
felt that the required use of the empirical equation without out
relief in the short period range would motivate practitioners to
revert to the more traditional, and apparently less conservative,
linear procedures. The current limitations are not founded directly
on theoretical principles or empirical data. Much of the reduction
in response of short period structures is due to soil-structure
interaction effects. In lieu of the capping of C1, FEMA 440 will
introduce adjustments to seismic demand intended to address
soil-structure interaction effects inelastic analyses [11]. These
are fundamentally similar to those in the NEHRP [5] intended for
linear analyses. Short, stiff buildings generally are more
sensitive to interaction between soil material strength and
stiffness with that of the structure and its foundations than are
longer period structures. This is patially accounted for by
modeling the stiffness and strength of foundation and supporting
soils in the structural analysis as outlined in FEMA 356 and ATC
40. FEMA 356 will include procedures to reduce spectral ordinates
for the kinematic effects of base slab averaging and embedment of
the structure (see Figure 14). Additionally, the document will
include procedures to modify the damping of the overall systems to
account for the inertia effects of foundation damping (see Figure
15). These improvements will rationally result in reductions in
estimated response for short period structures.
-
13
Figure 14:Reduction in free-field motion due to kinematic
effects of base slab averaging and
embedment. (from Stewart 2003)
Figure 15:Foundation damping
REFERENCES 1. BSSC, A Prestandard And Commentary For The Seismic
Rehabilitation Of Buildings, prepared
by the Building Seismic Safety Council; published by the Federal
Emergency Management Agency, FEMA 356 Report, 2000, Washington,
DC.
2. ATC, The Seismic Evaluation and Retrofit of Concrete
Buildings, Volume 1 and 2, ATC-40 Report, 1996, Applied Technology
Council, Redwood City, California.
3. Miranda, E. and Akkar S., “Evaluation of approximate methods
to estimate target displacements in nonlinear static procedures,”
PEER-2002/21, Proc. Fourth U.S.-Japan Workshop on Performance-Based
Earthquake Engineering Methodology for Reinforced Concrete
Building
1 1.5 2Period Lengthening, Teq/Teq
0
10
20
30
Fou
ndat
ion
Dam
ping
, βf (
%)
e/ru = 0PGA > 0.2g
PGA < 0.1g
h/rθ = 0.5
1.0
2.0
∼
0 0.2 0.4 0.6 0.8 1 1.2
Period (s)
0.4
0.5
0.6
0.7
0.8
0.9
1F
ound
atio
n/fr
ee-f
ield
RR
S
from
bas
e sl
ab a
vera
ging
(R
RS
bsa)
Simplified Modelbe = 65 ft
be = 130 ft
be = 200 ft
be = 330 ft
0 0.4 0.8 1.2 1.6 2
Period (s)
0
0.2
0.4
0.6
0.8
1
1.2
Fou
ndat
ion/
free
-fie
ld R
RS
from
em
bedm
ent e
ffect
s (R
RS
e)
Site Classes C and De = 10 ft
e = 20 ft
e = 30 ft
C
D
-
14
Structures, 22-24 October 2002, Toba, Japan, Pacific Earthquake
Engineering Research Center, University of California, Berkeley,
Dec. 2002, pages 75-86.
4. Ruiz-Garcia, J. and Miranda, E. “Inelastic displacement ratio
for evaluation of existing structures,” Earthquake Engineering and
Structural Dynamics. 32(8), 1237-1258, 2003.
5. Building Seismic Safety Council, BSSC. NEHRP Recommended
Provisions for Seismic Regulations for New Buildings and Other
Structures, Part 1 – Provisions and Part 2 – Commentary, Federal
Emergency Management Agency, Washington D.C., February, 2001.
6. Iwan, W. D. “Estimating inelastic response spectra from
elastic spectra.” Intl. J. Earthq. Engng. and Struc. Dyn., 1980, 8,
375-388.
7. Iwan, W. D. and Gates, N.C., “The effective period and
damping of a class of hysteretic structures,” Intl. J. Earthq.
Engng and Struc. Dyn., 1978, 7, 199-211.
8. Iwan, W. D., and Guyader, A.C., "An improved capacity
spectrum method employing statistically optimized linear
parameters," Paper No. 3020, 13th World Conference on Earthquake
Engineering, Vancouver, B.C., Canada, August 1-6, 2004
9. Guyader, A.C., and, Iwan, W.D., "A Statistical Approach to
Equivalent Linearization with Application to Performance-Based
Engineering,” California Institute of Technology, EERL Report No.
2004-04, 2004.
10. Aschheim, M., Tjhin, T., Comartin, C., Hamburger, R., and
Inel, M., "The scaled nonlinear dynamic procedure," ASCE Structures
Congress, Nashville, TN, May 22-26, 2004.
11. Stewart, J.P., Comartin, C.D., and Moehle, J.P.,
“Implementation of soil-structure interaction models in performance
based design procedures”, Paper No. 1546, 13th World Conference on
Earthquake Engineering, Vancouver, B.C., Canada, August 1-6,
2004
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