fema 440
TRANSCRIPT
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]
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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SSD
Strength and stiffness degrading model
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SSD
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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
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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For
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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
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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
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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
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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
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Strength and stiffness degrading model
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Strength and stiffness degrading model
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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
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Strength and stiffness degrading model
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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.
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
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
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 2
11
90
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
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 oT T
µµ
⎡ ⎤⎛ ⎞−⎢ ⎥− = −⎜ ⎟⎜ ⎟+ − −⎢ ⎥⎝ ⎠⎣ ⎦
(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