PEER2010 - 111 - ATC Modeling and Acceptance Criteria For Seismic Design An Analysis of Tall Buildings
PEER2010 - 111 - ATC Modeling and Acceptance Criteria For Seismic Design An Analysis of Tall Buildings
PEER2010 - 111 - ATC Modeling and Acceptance Criteria For Seismic Design An Analysis of Tall Buildings
RESEARCH CENTER
foundation
V
M
PEER 2010/111
(PEER/ATC-72-1)
OCTOBER 2010
PEER Report 2010/111
(also published as)
PEER/ATC-72-1
Prepared by
APPLIED TECHNOLOGY COUNCIL
201 Redwood Shores Pkwy, Suite 240
Redwood City, California 94065
www.ATCouncil.org
in collaboration with
Building Seismic Safety Council (BSSC)
National Institute of Building Sciences (NIBS)
Federal Emergency Management Agency (FEMA)
Prepared for
PACIFIC EARTHQUAKE ENGINEERING RESEARCH CENTER (PEER)
Jack P. Moehle, Principal Investigator
Stephen Mahin, Director
Yousef Bozorgnia, Executive Director
October 2010
Applied Technology Council Pacific Earthquake Engineering
Research Center
The Applied Technology Council (ATC) is a nonprofit, The Pacific Earthquake Engineering Research (PEER)
tax-exempt corporation established in 1973 through the Center, with headquarters at UC Berkeley, is a consortium
efforts of the Structural Engineers Association of of western U.S. universities working in partnership with
California. ATC’s mission is to develop state-of-the-art, business, industry, and government to identify and reduce
user-friendly engineering resources and applications for the risks from major earthquakes to life safety and to the
use in mitigating the effects of natural and other hazards economy.
on the built environment. ATC also identifies and
encourages needed research, and develops consensus PEER achieves its objectives of earthquake risk reduction
through a coordinated program of research, education, and
opinions on structural engineering issues in a non-
proprietary format, thereby fulfilling a unique role in partnerships with users of research results. PEER’s
funded information transfer. research program includes basic and applied components
of geology, seismology, various branches of engineering,
ATC is guided by a Board of Directors consisting of architecture, urban planning, and economics.
representatives appointed by the American Society of Civil
Engineers, the National Council of Structural Engineers PEER is supported by funding from the United States
Federal Government, State of California (various
Associations, the Structural Engineers Association of
California, the Western Council of Structural Engineers agencies), local governments, and the private sector.
Associations, and at-large representatives concerned with PEER Reports can be ordered at
the practice of structural engineering. Each director serves http://peer.berkeley.edu/publications/peer_reports.html
a three-year term.
or by contacting:
Project management and administration are carried out by
a full-time Executive Director and support staff. Project Pacific Earthquake Engineering Research Center
work is conducted by a diverse group of highly qualified 1301 South 46th Street
consulting professionals, thus incorporating the expertise Richmond, California 94804-4698
of individuals from academia, research, and professional Tel: 510-665-3448
practice who would otherwise not be available from any Fax: 510-665-3456
single organization. Funding for ATC projects is obtained Email: peer_editor@berkeley.edu
from government agencies and from the private sector in
the form of tax-deductible contributions. PEER Director: Stephen Mahin
PEER Executive Director: Yousef Bozorgnia
2010-2011 ATC Board of Directors
Ramon Gilsanz, President
Marc Levitan, Vice President
Bret Lizundia, Secretary/Treasurer
H. John Price, Past President
Cover Illustration: Courtesy of Joseph Maffei, Rutherford & Chekene, San Francisco, California
Preface
Using the workshop as a starting point, this report is the result of further
work under the PEER Tall Buildings Initiative to develop modeling
recommendations and acceptance criteria for design and analysis of tall
buildings. It is intended to serve as a resource document for the Guidelines
for Seismic Design of Tall Buildings, published as a companion report by
PEER (2010).
ATC also gratefully acknowledges Jack Moehle, Yousef Bozorgnia, and the
PEER Tall Buildings Project Advisory Committee for their input and
guidance in the completion of this report, Ayse Hortacsu and Peter N. Mork
for ATC report production services, and Charles H. Thornton as ATC Board
Contact.
iv Preface PEER/ATC-72-1
Table of Contents
Figure 2-24 Damping and drift demand data from buildings excited
by strong ground motions................................................. 2-41
Figure 3-14 Cyclic shear behavior of weak panel zone ....................... 3-24
Figure 3-17 Use of two springs to model trilinear behavior ................ 3-27
Figure 4-3 Biaxial fiber model for bending in two-dimensions ........... 4-4
Figure 4-4 Coupled model and results for a low-aspect ratio wall ...... 4-5
Figure 4-10 Impact of wall flexural strength on effective stiffness ..... 4-14
Figure 4-28 Influence of mesh size on wall strain distribution ............ 4-28
Figure 4-29 Coupling beam effective flexural stiffness ratios. ............ 4-32
Figure 4-33 Comparison of: (a) effective stiffness; and (b) backbone
relations for coupling beam test results ............................ 4-34
Figure 4-38 Configuration and plan section of tall core wall building
system used in parametric studies .................................... 4-40
Figure 4-39 Variation in shear force over height in the: (a) north-
south direction; and (b) east-west direction, for each
case of relative stiffness ................................................... 4-42
Figure 4-52 Floor plan and simplified model of the combined slab-
column frame and core wall system ................................. 4-54
Figure 4-53 Application of effective width model to core wall .......... 4-54
1.1 Background
Chapter 1 provides background information and context for the overall PEER
Tall Buildings Initiative.
Nonlinear response history analysis is the best tool currently available for
predicting building response at varying levels of ground motion intensity.
Various aspects of nonlinear analysis, such as acceptance criteria, element
discretization, and assumptions on modeling of energy dissipation through
viscous damping, must be tailored to the specific features of the analytical
representation of the system, and the extent to which various behavioral
effects will be captured in the nonlinear component models.
In between the two extremes are distributed inelasticity (fiber) models, which
capture some aspects of behavior implicitly, such as integration of flexural
stresses and strains through the cross section and along the member, and
other effects explicitly, such as definition of effective stress-strain response
of concrete as a function of confinement. These models typically enforce
some behavior assumptions (e.g., plane sections remain plane) in
combination with explicit modeling of uniaxial material response.
Figure 2-2 illustrates the key features of an inelastic hinge model for
reinforced concrete beam-column elements. The features of this element are
generally applicable to other types of elements. This example is taken from a
study of reinforced concrete columns by Haselton et al. (2008) making use of
a degrading cyclic model developed by Ibarra and Krawinkler (2005). In this
example, inelastic response is idealized by a backbone curve (Figure 2-2b)
that relates moment to rotation in the concentrated hinges. The definition of
the backbone curve and its associated parameters depend on the specific
attributes of the nonlinear model used to simulate the hysteretic cyclic
response (Figure 2-2c). The following important features of this model will
be highlighted in later sections of this report:
The backbone curve is generally expected to capture both hardening and
post-peak softening response. The peak point of the curve is sometimes
referred to as the “capping point,” and the associated deformation
capacity is the “capping deformation.” The extent to which cyclic
deterioration is modeled in the analysis will determine the extent to
which the backbone curve is calibrated to initial or degraded component
Cap,pl
Exp. Results
Model Prediction
p Shear Force (kN) 200
100
My
0
Ke Kpc
-100
-200
y -300
-0.1 -0.05 0 0.05 0.1
Column Drift (displacement/height)
Unlike linear analyses, nonlinear analyses are load path dependent, in which
the results depend on the combined gravity and lateral load effects. For
seismic performance assessment using nonlinear analysis, the gravity load
applied in the analysis should be equal to the expected gravity load, which is
different from factored gravity loads assumed in standard design checks.
In general, the expected gravity load is equal to the unfactored dead load and
some fraction of the design live load. The dead load should include the
structure self weight, architectural finishes (partitions, exterior wall, floor
and ceiling finishes), and mechanical and electrical services and equipment.
The live load should be reduced from the nominal design live load to reflect:
(1) the low probability of the nominal live load occurring throughout the
building; and (2) the low probability of the nominal live load and earthquake
occurring simultaneously. Generally, the first of these two effects can be
considered by applying a live load reduction multiplier of 0.4 and the second
by applying a load factor of 0.5 (such as is applied to evaluation of other
extreme events).
The net result is a load factor of 0.4 x 0.5 = 0.2, which should be applied to
the nominal live load. For example, in a residential occupancy with a
where D is the nominal dead load, and L is the nominal live load. In the case
of storage loads, only the 0.5 factor would apply, and the net load factor on
storage live loads should be taken as 0.5. Expected gravity loads should also
be used as the basis for establishing the seismic mass to be applied in the
nonlinear analysis.
Vertical gravity loads acting on the entire structure, not just the seismic-
force-resisting elements, should be included in the analysis in order to
capture destabilizing P-Delta effects. Nonlinear analysis should include
leaning columns with applied gravity loads that rely on the seismic-force-
resisting system for lateral stability.
In this report, the emphasis is on defining capacities for two structural limit
states: (1) the onset of structural damage requiring repair; and (2) the onset of
significant degradation in structural components. The onset of structural
damage requiring repair is envisioned as one of several possible metrics for
assessing direct economic losses and disruption of building functionality.
Initiation of structural damage also corresponds to the point at which elastic
analysis may no longer be adequate for assessing performance.
2.2 Deterioration
Collapse implies that the structural system is no longer able to maintain its
gravity load-carrying ability in the presence of seismic effects. Local
collapse may occur, for instance, if a vertical load-carrying component fails
in compression, or if shear transfer is lost between horizontal and vertical
components (e.g., punching shear failure between a flat slab and a column).
Global collapse occurs if a local failure propagates, or if an individual story
16.00 16.00
3 12.00
12.00
1 7,9
5
7 5
9 3
8.00 1
8.00
39
41
TIP LOAD (kips)
4.00 4.00
43
TIP LOAD (kips)
15
0.00 0.00
-4.00
-4.00
42
40
8 -8.00
-8.00 2
6 4
4 6
2
-12.00 -12.00
-16.00
-16.00
-4.00 -2.00 0.00 -2.00 4.00
-2.00 -1.00 0.00 1.00 2.00
(a) (b)
Figure 2-3 Plots showing different rates of deterioration: (a) slow
deterioration; and (b) rapid deterioration (ATC, 1992).
1
2
3
3
The first three modes of cyclic deterioration are observed in the cyclic
response of all structural components. The fourth mode (accelerated
reloading stiffness deterioration) is not discernible in components with
behavior that is controlled by flexure, and is represented by “fat” hysteresis
loops (Figure 2-5).
4
x 10 Uang-SSR-00-LS-1-MomentRotation 4 Uang-SSR-00-LS-2-MomentRotation
2 x 10
2
1.5 1.5
1 1
Moment (k-in)
Moment (k-in)
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-1.5 -1.5
-2 -2
-0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1
Chord Rotation (rad) Chord Rotation (rad)
(a) (b)
Figure 2-5 Hysteretic response of identical steel beam specimens under different loading
histories (Uang et al., 2000).
For the case illustrated in Figure 2-5a, it is also observed that the hysteresis
loops stabilize at large inelastic cycles, which indicates that there is a residual
In Figure 2-6a, the stable drift capacity of the column appears to be at least
8%, but in Figure 2-6b cyclic deterioration appears to set in around 4% drift.
The stable monotonic drift capacity is quite possibly larger than 8% because
the load was reversed at 8% even though no decrease in strength was evident.
200 200
150 150
100 100
(kN)(kN)
(kN)(kN)
50 50
ForceLoad
Load
0 0
Force
Lateral
Lateral
-50 -50
-100 -100
-150 -150
-200 -200
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
Drift (%) Drift (%)
(a) (b)
Figure 2-6 Hysteretic response of identical reinforced concrete column specimens under
different loading histories (Kawashima, 2007).
Figure 2-7 shows two incremental dynamic analysis (IDA) curves showing
spectral acceleration at the first mode period of the structure versus computed
maximum story drift, for one specific ground motion and one specific frame
structure. One curve was obtained from analysis with non-deteriorating
structural component models (i.e., the hysteretic response is assumed to be
bilinear and no monotonic or cyclic deterioration modes are considered). Since
P-Delta effects are not large enough to overcome the strain-hardening effects
inherent in the component models, the IDA curve continues to rise to large drifts
of more than 20%, until the analysis was stopped.
The second curve was obtained from analysis with deteriorating component
models. At relatively small drifts, the responses are identical, but the curves
diverge once cumulative damage sets in. The slope of the second IDA curve
decreases rapidly between Sa(T1) of 2.5g and 2.8g, where it approaches zero. At
this point, a small increase in intensity causes a large increase in story drift,
which indicates dynamic instability in the analytical model. Presuming that the
model is accurate, this implies sidesway collapse of a single story or a series of
stories in the structural system. The ground motion intensity level associated
with dynamic instability can be denoted as the collapse capacity of the specific
structure, given the specific ground motion.
4
Non-deteriorating system
Deteriorating system
Collapse 3
Capacity
S a (T 1 )/g
0
0 0.02 0.04 0.06 0.08
max. interstory drift ratio
Figure 2-7 Incremental dynamic analysis (IDA) curves for a moment-resisting
frame example using non-deteriorating and deteriorating
component models.
The following concepts are specifically described for rotational springs that
represent plastic hinge regions. They can be applied equally well to
translational springs that represent shear force-deformation modes, and can
be adapted to other localized or component or element force-deformation
modes built into a structural analysis program.
Backbone Curve
The initial backbone curve is close to, but is not necessarily identical to, the
monotonic loading curve. It usually contains compromises that are made in
order to simplify response description. For instance, it might account for an
average effect of cyclic hardening (which is likely small for reinforced
concrete components, but can be significant for steel components). In
concept, the differences between the initial backbone curve and the
monotonic loading curve are small, and the terms initial and monotonic are
interchangeable for practical purposes.
Fc
Fy
Fr
Ke
y c r u
p pc
The properties of the initial (monotonic) backbone curve in the positive and
negative directions can be different, as necessitated, for instance, by the
presence of a slab on a steel beam or unequal reinforcement in a reinforced
concrete beam. There might also be additional considerations that affect the
construction of a backbone curve. If the initial stiffness is very different
from the effective elastic stiffness, response can be affected, even close to
collapse, and initial stiffness should become part of the modeling effort.
Residual strength (Fr in Figure 2-8) may or may not be present. Residual
strength is present in most steel components, unless fracture occurs before
the component strength stabilizes at a residual value. The ultimate
deformation capacity usually is associated with a sudden, catastrophic failure
mode. In steel components, this can be ductile tearing associated with severe
local buckling, or fracture at weldments. It is possible that the ultimate
deformation capacity, u, is smaller than the deformation at which a residual
strength is reached, r.
As defined here, the initial backbone curve presumes that cyclic deterioration
will be incorporated in the analytical component model. If this is not
Fy+ F 1 F 1
2,7 Fmax F 2
2,6 Fy+ 8 Fy+ 1
5
Ke Krel,b
Ke Krel Ke 8
Figure 2-10 Simulations obtained with a modified Bouc-Wen model (Foliente, 1995).
F Ks,0 2 Fref,0+ F
Fy+
1 Fref,1+ 2
F1+ 8 Ks,1+ 1
Fy+
Ke 7
8
- Ke
c1- c0
c1- 6 3
7 0 3 co+ c1+
-
c0 0 c1+ c0+
Krel
Fy+
F 2 F 1 2
t1+
1 Fy+ 7 8 9
Interruption Ke 7
(disregard stiffness det) Ke
Ku,1
Ke
6 0 c0- 3
3 c0+
6 0
Ku,2 Krel
Original
Envelope
Fy- 4
4 5 Fy-
5 t1-
4 Engelhardt-E9608-UTDB3-MomentRotation Nagai-Kajima-RB-15-Specimen-BL-22-MomentRotation
x 10 1000
4
Ke = 2250000 Ke = 145450
+ 800
M = 28500 +
M y = 800
3 y-
M y = -28500 -
M y = -800
p = 0.020
600 +
p = 0.057
2 = 0.45 -
pc 400 p = 0.057
= 1.5 +
pc = 0.140
s
1 = 1.5 -
Moment (k-in)
pc = 0.140
Moment (k-in)
c 200
= 1.5 s = 0.3
a
0 = 1.3 0 c = 0.3
k
M c/M y = 1.05 a = 0.3
-200
-1 k = 0.3
+
-400 M c/M y = 1.03
-
-2 M c/M y = 1.03
-600
-3
-800
-4 -1000
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Chord Rotation (rad) Chord Rotation (rad)
(a) (b)
Figure 2-13 Ibarra-Krawinkler model calibration examples: (a) steel beam; and (b) reinforced
concrete beam.
The need to simulate structural response far into the inelastic range has led to
the development of other versatile models, such as the smooth hysteretic
degrading model developed by Sivaselvan and Reinhorn (2000), which
includes rules for stiffness and strength deterioration as well as pinching. An
example of a simulation using the Sivaselvan-Reinhorn model is shown in
Figure 2-14.
200.0
150.0
100.0
50.0
0.0
-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
-50.0
-100.0
-150.0
-200.0
Displacement
(a) (b)
Figure 2-14 Example of a simulation using the Sivalsevan-Reinhorn model showing: (a)
experimental results; and (b) calibrated simulation (SAC, 1999b).
(a) (b)
(c)
Figure 2-15 Song-Pincheira model: (a) backbone curve; (b) hysteresis rules for cycles of increasing
deflection amplitude; (c) hysteresis rules for small amplitude or internal cycles (Song
and Pincheira, 2000).
Backbone curve
Cyclic skeleton curve
0.8Fc-
Load Fc-
Displacement
(a) (b)
Figure 2-16 Monotonic and cyclic responses of identical specimens, and skeleton curve fit to
cyclic response for: (a) steel beam (Tremblay et al., 1997); and (b) plywood shear
wall panel (Gatto and Uang, 2002).
Note that Option 2 and Option 3 are similar in concept, except that in
Option 2, the cyclic skeleton curve is based on test data, while in Option
3 it is based on factors applied to the initial backbone curve.
Option 4 – no strength deterioration in analytical model. If the post-
capping (negative tangent stiffness) portion of a modified backbone
curve is not incorporated in the analytical model (i.e., a non-deteriorating
model is employed), then the ultimate deformation of a component
should be limited to the deformation associated with 80% of the capping
strength on the descending branch of the modified backbone curve, as
obtained using Option 2 or Option 3. No credit should be given for
undefined strength characteristics beyond this deformation limit in the
analysis (see Figure 2-17d).
Monotonic Modified
backbone curve backbone curve
u
(a) Option 1 – cyclic deterioration in analytical model (b) Option 2 – cyclic envelope (skeleton) curve
Fc
0.8Fc
Modified Modified
backbone curve backbone curve
u u
(c) Option 3 – factored initial backbone curve (d) Option 4 – no strength deterioration
In Figure 2-17, it can be seen that the greater the simplification in component
modeling, the more the inelastic deformation capacity is reduced. This is
most evident in Figure 2-17d, in which the attainment of the estimated u
severely limits the inelastic deformation capacity.
Plastic Hinge Rotation Capacity Effect on c (MRF) Post-Cap. Rotation Capacity Effect on c (MRF)
N = 8, T1 = 1.2, = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, = 0.05 N = 8, T1 = 1.2, = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, = 0.05
p = var., pc/p = 5.0, = 20, Mc/My = 1.1 p = 0.03, pc/p = var., = 20, Mc/My = 1.1
3 3
= 0.33 = 0.33
= 0.08 = 0.08
2 2
c (Sa/g)
c (Sa/g)
1.5 1.5
1 1
0.5 0.5
0 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 5 10 15
Plastic hinge rotation capacity p Post-capping rotation capacity ratio pc/p
(a) Effect of pre-capping plastic rotation p, MRF (b) Effect of post-capping rotation range pc, MRF
Cyclic Deterioration Effect on c (MRF) Plastic Hinge Rotation Capacity Effect on c (SW)
N = 8, T1 = 1.2, = var.,Stiff.&Str. = Shear, SCB = 2.4-1.2, = 0.05 N = 8, T1 = 0.80, = var.,Str. = Unf., = 0.05
p = 0.03, pc/p = 5.0, = var., Mc/My = 1.1 p = var., pc/p = 1.0, = 20, Mc/My = 1.1
3 6
= 0.33 = 0.50
1.5 3
2
1
1
0.5
0
0
0 10 20 30 40 50 60 0.00 0.01 0.02 0.03 0.04
Cyclic deterioration parameter Plastic hinge rotation capacity p
(c) Effect of cyclic deterioration parameter , MRF (d) Effect of pre-capping plastic rotation p, SW
Figure 2-18 Effects of deterioration parameters on median collapse capacity of generic 8-story
moment-resisting frame (MRF) and shear wall (SW) structures (Zareian, 2006).
Figures 2-18a, 2-18b, and 2-18c are for generic 3-bay moment-resisting
frames, and Figure 2-18d is for generic shear wall structures that develop a
flexural hinge at the base. Except for the moment frame system with low
base shear strength ( = 0.08), the results show a clear dependence on the
deterioration parameters. The exception in the case of low base shear
200
100
Force
-100
-200
-0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60
Displacement
(Sac/g)
1.5 3
1 2
0.5 1
0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6
(a) (b)
Figure 2-21 Effects of P-Delta on median collapse capacity (Sac/g) of: (a) 8-story moment-
resisting frame; and (b) shear wall structure deforming in a flexural mode
(Zareian, 2006).
0.1
0.08
0.06
0.04
0.02
0
0 0.01 0.02 0.03 0.04 0.05
Roof Drift Angle
Figure 2-22 Base shear versus roof displacement pushover curves for the SAC 20-story Los
Angeles structure (FEMA, 2000b).
0.10 1.2
Story Drift Angle
Scale Factor
0.05 0.8
-0.05 0
0 10 20 30 40 50 0 0.05 0.1 0.15 0.2 0.25
Time (seconds) Maximum Story Drift Angle
(a) (b)
Figure 2-23 Dynamic response of SAC 20-story Los Angeles structure using four different analytical
models shown as: (a) response histories; and (b) incremental dynamic analyses (FEMA,
2000b).
2.4 Damping
The extent to which all of the energy dissipation in these elements is captured
through hysteretic response in the nonlinear analysis depends on the specific
characteristics of the model. For example, concentrated plasticity (discrete
hinge) models in beams and columns may not capture energy dissipated by
gradual steel yielding or concrete cracking prior to the formation of a hinge.
On the other hand, fiber-type analyses will do a better job of capturing
energy dissipation at lower values of deformation, but may still not capture
all the sources of energy dissipation (such as through reinforcing bar bond
deterioration and bolt slip). Since it is common practice to mix elastic and
nonlinear elements (e.g., modeling the lower hinging portion of shear walls
with nonlinear fiber-type analysis and the upper portions elastically), energy
dissipation associated with yielding and cracking in the portions of the
structure that are modeled with elastic elements will not be captured.
For practical reasons, there are many structural components that are not
explicitly modeled in the analysis, but are expected to undergo inelastic
deformations. Chief among these are components of the gravity system,
including floor slabs, gravity beams, gravity columns, and their associated
connections. Yielding and cracking of gravity components and connections
caused by imposed lateral deformations is a source of energy dissipation in
the system that should be implicitly accounted for in the analysis when not
explicitly included in the model.
Given that tall buildings can experience significant story drifts due to service
wind load, interior partitions, curtain walls, and mechanical and electrical
risers in tall buildings are usually detailed to minimize their interaction with
the structure. Accordingly, these components will tend to contribute less
damping in tall buildings than they provide in low-rise buildings. However,
the contribution of each will tend to vary from building to building
depending on the architectural layout (e.g., amount of interior walls per floor
area) and the method of attachment to the structure.
Wind design criteria for tall buildings generally consider two limit states: one
associated with occupant comfort and a second associated with structural
safety. The designs are often based on wind-tunnel studies, with two levels
of equivalent viscous damping assumed for each limit state. Given that
damping is amplitude dependent, smaller values are typically specified for
serviceability (occupancy comfort) than for safety (structural member
design).
Only recently has nonlinear response history analysis been regularly applied
in structural engineering design of buildings to resist earthquakes. As a
result, many of the currently available guidelines on damping are intended
for use with elastic dynamic analysis. For example, the Los Angeles Tall
Buildings Structural Design Council (LATBSDC) seismic design
requirements for tall buildings specify equivalent critical viscous damping
values of 5% for steel framed buildings and 10% for reinforced concrete
buildings for analysis at design-level ground motions, and 7.5% for steel and
12% for reinforced concrete buildings for analysis at MCE-level ground
motions (Harder, 1989; Martin and Harder, 1989). Since these damping
values are intended to account for inelastic (hysteretic) effects within the
context of elastic response history analysis, they are inappropriate and should
not be used with nonlinear dynamic analyses.
With the advent of lower cost systems for building instrumentation and
monitoring, data on measured response of buildings are becoming
increasingly available. Much of the data are limited to small amplitude
vibrations, typically excited by wind, mechanical shakers, or small
earthquakes. There are some data available from buildings subjected to
On the other hand, many studies of buildings in the United States subjected
to strong ground motions have utilized spectral system identification
techniques to calculate the damping (Celebi, 1998), which may tend to
overestimate damping effects. Jeary (1986) provides further discussion on
methods that are commonly used to quantify damping, and emphasizes the
inherent limitations of certain methods, and the errors that can arise when
methods are misapplied. These differences should be considered when
attempting to calculate small damping values from low amplitude vibrations
from random loading events.
Goel and Chopra (1997) compiled and analyzed recorded strong motion data
for 85 buildings that were subjected to strong ground motions from eight
earthquakes in California, from the 1971 San Fernando earthquake through
the 1994 Northridge earthquake. With the primary purpose of examining
16
14
12
% Damping
10
8 Longitudinal
6 Transverse
0
0 10 20 30 40 50 60 70
Number of Stories
0.7
0.6
Roof Drift Ratio (%)
0.5
0.4
Longitudinal
0.3
Transverse
0.2
0.1
0
0 10 20 30 40 50 60 70
Number of Stories
16
14
12
% Damping
10
8 Longitudinal
6 Transverse
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Roof Drift Ratio (%)
To examine this further, roof drift ratios are plotted versus the number of
stories in Figure 2-24b, and the damping ratio is plotted versus roof drift ratio
in Figure 2-24c. Both plots suggest that there are no discernable trends
between drift demand and story height that play a major role in the variation
of observed damping.
Data reported by Goel and Chopra reflect similar findings in other studies of
measured strong motion response in the United States over the past thirty
years, including Bradford et al. (2004), Celebi (1998), Celebi (2006), Hudson
and Housner (1954), Li and Mau (1997), Maragakis et al. (1993), Rodgers
and Celebi (2006), Skolnik et al. (2006), Stephen et al. (1985) and Trifunac
(1970).
Damping data are also available from vibration tests of real buildings, where
vibration is typically induced by a mechanical shaker. Since forcing
functions are known, and tests can be repeated multiple times, force vibration
tests have the advantage that data reporting and analysis are more
comprehensive. However, displacement amplitudes in forced vibration tests
are typically much smaller, so allowances are needed when comparing
damping data from forced-vibration studies to data recorded from strong
earthquake shaking.
Figure 2-25 shows measured damping data reported by Satake et al. (2003)
for buildings in Japan. Data are reported for 127 steel frame buildings up to
280 meters (70 stories) in height, and 68 buildings of reinforced concrete or
mixed steel/concrete construction up to 170 meters (45 stories) in height.
Most of the steel frame buildings are office or hotel occupancies, and most of
the reinforced concrete and mixed steel/concrete buildings are residential
occupancies. About half of the data are from forced-vibration testing, and
half are from micro-tremor or wind-induced vibration testing. A few of the
Figure 2-25 Measured damping from buildings in Japan (Satake et al., 2003).
The trend toward lower damping with increasing building height in Figure
2-25 is similar to the Goel and Chopra data in Figure 2-24, although in
absolute terms, the damping values are lower. In Figure 2-25, the range of
damping is from about 0.5% to 8% of critical damping, versus 1% to 15% of
critical damping in Figure 2-24. The smaller values in Figure 2-25 are most
likely due to the differences in the vibration displacement amplitudes
between the two studies. Assuming a demarcation of 30 stories between
low/mid-rise and high-rise buildings (roughly equivalent 120 meters), the
maximum critical damping ratios plotted in Figure 2-25 range up to 4% (for
steel) and 8% (for reinforced concrete) in low/mid-rise buildings, and up to
2% in high-rise buildings.
Also shown in Figure 2-25 is a simple regression formula that relates critical
damping to the inverse of building height. Satake et al. (2003) attribute most
of the change in damping with building height to the decreased significance
of foundation and soil damping effects in taller buildings. They also examine
other trends in the data and note that the damping ratios are slightly larger in
apartment buildings than office buildings, which they suggest is related to the
density of interior partitions.
Measurements taken from buildings in strong wind storms are another source
of data on damping effects. As buildings taller than 30 stories are a primary
It should be noted, however, that even the largest recorded amplitudes in the
high amplitude plateau are on the order of 0.02% roof drift. This is well
below the amplitudes associated with serviceability or safety limit states for
strong ground motions (i.e., drifts on the order of yield-level drifts of 0.5% to
1%). Unfortunately, there are no studies relating damping at the high
amplitude plateau for wind loading to damping at larger drifts expected under
earthquake shaking.
Table 2-1 Selected Results of Measured Damping in Tall Buildings under Wind-
Induced Vibration
Damping
Building Description (% critical) Reference
57-story steel frame office building 0.8% Kijewski-Correa et al.
(2007), case B1
73-story reinforced concrete shear wall 0.8% to 1.2% Kijewski-Correa et al.
with outrigger frames (2007), case S1
>50-story steel perimeter tube system 0.9% Kijewski-Correa et al.
(2007), case C1
>50-story reinforced concrete shear wall 1.4% Kijewski-Correa et al.
with frames (2007), case C2
>50-story steel frame tube system 1.0% Kijewski-Correa et al.
(2007), case C2
79-story reinforced concrete core with 0.4% Li et al. (2002)
outriggers
70-story composite braced frame 0.5% Li et al. (2000)
Measured damping data from various shaking table tests are summarized in
Table 2-2. The table includes data for reduced scale (1/3 to 1/2 scale) tests of
reinforced concrete frame (or frame-wall) systems and steel braced frame
systems. Data are reported in terms of the percentage of critical damping in
the first mode. In the initial or undamaged condition, damping in reinforced
concrete frames ranges from 1% to 3% of critical. In structures that have
Table 2-2 Measured Damping versus Level of Damage from Shaking Table Tests
Measured Damping (% critical)
Test Description versus Level of Damage Reference
RC Frames (2) Undamaged: 1.4% to 1.9% Elwood and
1-story, 3-bay Yielded: 2.1% to 3.7% Moehle (2003)
(1/2 scale) Significant: 3.9% to 5.4%
RC Wall-Frame Undamaged: 1.9% to 2.2% Aktan et al. (1983);
7-story Slight: 3.5% to 3.7% Bertero et al.
(1/5 scale) Significant: 6.9% to 7.5% (1984)
RC Flat Plate-Frame Undamaged: 1.2% to 1.7% (negligible drift) Diebold and
2-story, 3-bay Slight: 2.4% to 2.6% (0.002 to 0.011 drift) Moehle (1984)
(1/3 scale) Moderate: 5.0% (0.017 to 0.034 drift)
Significant: 7.2% (0.053 drift)
RC Frame Undamaged: 1.9 to 2.2% Oliva (1980)
2-story, 1-bay Damaged: 3.9 to 5.3%
(1/3 scale)
RC Frame Undamaged: 2.7% to 3.7% (0.001 to 0.003 drift) Shahrooz and
3- to 6-story, 2-bay Moderate: 4.9% to 6.4% (0.012 drift) Moehle, (1987)
(1/3 scale) Significant: 9.6% to 11.1% (0.015 to 0.02 drift)
RC Frames (12) Undamaged: 1.4% to 2.9% Shin and Moehle
1-story, 3-bay (2.1% avg., 0.31 COV) (2007)
(1/3 scale)
RC Frame Undamaged: 1.9% Moehle et al.
3-story, 3-bay, (2006)
(1/3 scale)
Steel EBF Undamaged: 0.7% Whittaker et al.
1 bay, 6-story (1987)
(1/3 scale)
Steel CBF Undamaged: 0.7% to 1.3% Whittaker et al.
1 bay, 6-story (1988)
(1/3 scale)
Damping effects measured in shaking table tests can also be inferred from
comparisons with nonlinear analyses of the tests. For example, nonlinear
analyses with 2% viscous damping resulted in accurate comparisons to the
shake table tests by Shin and Moehle (2007). For shaking table tests of a
reinforced concrete bridge pier, Petrini et al. (2008) compared various
viscous damping assumptions made using fiber-type and plastic hinge
The quantification and definition of damping are integrally linked with how
damping is modeled. For elastic analyses, damping is defined in terms of
equivalent viscous damping through the velocity dependent term, [C], in the
equation of motion, as follows:
This is done for mathematical convenience, since the velocity is out of phase
with displacement and acceleration, and thus provides an easy way to
incorporate a counteracting force to damp out motions in a linear analysis.
To facilitate modal analyses, the damping matrix is often defined using either
the classical Rayleigh damping assumption, where [C] is calculated as a
linear combination of the mass [M] and stiffness [K] matrices, or modal
damping, where [C] is a combination of specified damping amounts for
specific vibration modes (usually elastic vibration modes). These damping
formulations are explained below.
[C ] aM [ M ] aK [ K ] (2-4)
1 Ti T j
a M 4 , and a K (2-6)
Ti T j Ti T j
Figure 2-27 Variation in percent of critical damping for mass, stiffness, and
Rayleigh proportional damping with = 2% at T1 = 5 seconds.
1
ci
1
[C ] T (2-7)
Where [] is the matrix of eigenvectors (mode shapes) and [ci] is a diagonal
matrix of damping coefficients for each mode. Following Chopra (2007),
Equation 2-7 can be implemented through the following equation:
N 4n T
C M n n
M (2-8)
n1 TM
n n
Where [M] is the diagonal mass matrix, n, n, and n are the percent critical
damping, period, and eigenvector (mode shape) for mode n, Mn (=nT[M]n)
is the generalized mass for mode n, and N is the number of modes included
in the calculation. In elastic analyses, a key advantage of modal damping
over Rayleigh or mass/stiffness-proportional damping is that the target
damping amounts in each vibration mode can be independently specified. It
is not clear whether or not this attribute of modal damping has the same
benefits in nonlinear analyses, where the vibration modes are not uniquely
defined.
Unlike linear analysis, where the elastic stiffness and percentage of critical
modal damping remain constant, in nonlinear analysis the stiffness matrix
softens due to inelastic effects, and the relative significance of damping can
change dramatically during the analysis. For example, consider a case where
the damping matrix, [C], is defined based on the initial elastic stiffness and
the fundamental period of vibration. If the damping matrix is fixed during
the analysis, then as the structure softens and the effective first-mode period
elongates, the percentage of damping in the elongated fundamental mode will
tend to increase. This increase can be reasoned from the equation of motion
(Equation 2-3), where the relative significance of the damping matrix would
Charney (2006) shows how the third option provides the most insurance
against developing excessive damping forces, although there is not a clear
physical basis for reducing the damping matrix to maintain a fixed critical
damping percentage for pseudo-vibration modes based on the tangent
stiffness. There are differing views on which of these options are most
appropriate. A number of researchers, such as Charney (2006) and Petrini et
al. (2008), have advocated the second option (i.e., fixing proportionality
constants applied to the updated tangent stiffness matrix) as a practical way
to avoid excessive damping in inelastic analysis, but others have countered
that the first option (i.e., constant damping) is legitimate.
To illustrate the issues, Powell (2008) examined how the choice of damping
formulation influences the effective damping in structures where the initial
period has elongated. Table 2-3 summarizes illustrative data from this
analysis, where effective damping coefficients are compared for a structure
with an initial elastic period of T1 = 1 second, a higher mode period Thigher =
6 Modal 4% 6% 4% 6%
A related question is the extent to which variability affects both the demand
and capacity side of the equation. In elastic analyses, it is relatively
straightforward to separate the demand and capacity effects. In nonlinear
analyses, demand and capacity are coupled. Variability in plastic hinge
properties will affect both the calculated demands and the rotation limits that
are used to assess the response.
Fully restrained connections are those in which the relative rotation between
the beam and the column is small in comparison with joint rotations and
panel zone distortions. In such connections, the rotation capacity can be
limited by localized fractures from a variety of sources, including:
Inadequate weld material quality or workmanship,
Inadequate ductility of base material,
Early cracking at web copes (weld access holes),
Stress and strain concentrations in the beam flanges at weldments,
Excessive shear deformations in joint panel zones,
Bolt slippage in the web shear tab connection, and
Net section fracture in the case of cover-plated bolted connections.
The same is true for the many finite element models that have been
developed and published in the literature. At this time this is not quite
feasible, thus, the only practical solution is to develop empirical rules for
adjusting of the backbone curve and cyclic deterioration parameters for local
and lateral-torsional buckling. Data for this purpose are discussed in Lignos
and Krawinkler (2009).
Most steel elements with hysteretic behavior that deteriorates due to local
instabilities approach stabilization of the hysteretic response at large inelastic
deformations. This stabilization occurs at a residual strength level that is
some fraction of the yield strength. It happens when stresses originally
carried in the buckled portions of the cross section have redistributed to
unbuckled portions of the cross section. While full stabilization rarely
occurs, it is observed that the rate of deterioration becomes small enough to
be neglected in analytical modeling.
At very large plastic rotations, cracks can develop in the steel base material
at the apex of the most severe local buckle, followed by rapid crack
propagation, ductile tearing, and essentially complete loss of flexural
strength. The deformation associated with this rapid strength loss is termed
the ultimate plastic rotation.
Steel beams are often part of a composite slab system. The presence of a
composite slab will move the neutral axis, change the moment-rotation
relationship, and affect the bending strength in both the positive and negative
directions (Ricles et al., 2004). This effect is not captured in tests of bare
steel connection subassemblies. In the positive moment direction (top flange
in compression), the presence of a slab will delay local instabilities but will
cause higher tensile strain demands in the bottom flange and welds. In the
If the slab is thick, or the beam depth is small, this increase in strength can be
a dominant factor. In the example shown in Figure 3-1, the capping rotation
is unsymmetric in the two loading directions (about 3% in positive bending
versus 1.2% in negative bending).
Figure 3-1 Hysteretic response of a steel beam with composite slab (data
from Ricles et al., 2004).
In an actual system, the floor slab and adjacent columns will restrain axial
contraction or expansion of the top and bottom flanges of a steel beam,
reducing the effect of local instabilities on rotation capacities. In most
experimental tests, however, the beam is free to contract or expand in the
axial direction. Axial restraint that is provided by the slab and adjacent
columns will have a positive effect on rotation capacity that is not captured in
currently available experimental data. The extent to which this effect helps
in increasing rotation capacity is unknown, but estimated to be significant.
Additional testing is necessary to reliably quantify this effect for use in
nonlinear analysis.
This section describes the basis for quantification of steel beam parameters
including the pre-capping plastic rotation, p, and post-capping rotation, pc,
for the initial moment-rotation backbone curve defined in Figure 2-8, as well
as the reference cumulative plastic rotation parameter, defined as part of
Equation 2-2 for calculating the hysteretic energy dissipation parameter (for
cyclic deterioration). These parameters are needed to develop a moment-
rotation model that explicitly incorporates cyclic deterioration, (modeling
Option 1 presented in Section 2.2.5).
Experimental data from steel beam tests have been collected in several
databases (PEER, 2007; Kawashima, 2007; Lignos and Krawinkler, 2007;
Lignos and Krawinkler, 2009). These data were used to quantify properties
for nonlinear modeling of steel beams.
The following four data sets were used to obtain statistical information on
trends between modeling parameters and selected geometric properties:
All tests on beams with non-RBS connections, for beam depths, d,
ranging from 4 inches to 36 inches, except tests with early fracture
problems
All tests on beams with RBS connections, for beam depths, d, ranging
from 18 inches to 36 inches
Applicable tests on beams with non-RBS connections, for beam depths, d
≥ 21 inches
Tests on beams with RBS connections, for beam depths, d ≥ 21 inches
All specimens in each data set are represented, spanning a wide range of
geometric and material properties. The curves shown represent log-normal
distributions fitted to the data points. The plots reveal statistical
characteristics, but do not display dependencies on individual properties.
Each plot shows CDFs for RBS and non-RBS (“other than RBS”)
connections. Results for the four data sets are comparable, but, in general,
median values of parameters for beams with non-RBS connections are
smaller (and the dispersion is larger) than for beams with RBS connections.
CDFs for Plastic Rotation Capacity p (All Data) CDFs for Plastic Rotation Capacity p (d>=21")
1 1
Probability of Exceedence
Probability of Exceedence
0.8 0.8
0.6 0.6
0.4 0.4
p(RBS) p(RBS)
Fitted Lognormal, =0.33 Fitted Lognormal, =0.29
0.2 0.2
p(Other than RBS) p(Other than RBS)
Fitted Lognormal, =0.43 Fitted Lognormal, =0.31
0 0
0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08
p p
(a) (b)
Figure 3-3 Cumulative distribution functions for pre-capping plastic rotation, p, for: (a) full data sets;
and (b) beam depths, d ≥ 21 in.
Probability of Exceedence
0.8 0.8
0.6 0.6
0.4 0.4
pc(RBS) pc(RBS)
(a) (b)
Figure 3-4 Cumulative distribution functions for post-capping rotation, pc, for: (a) full data sets; and
(b) beam depths, d ≥ 21 in.
Probability of Exceedence
0.8 0.8
0.6 0.6
0.4 0.4
(RBS) (RBS)
0.2 Fitted Lognormal, =0.34 0.2 Fitted Lognormal, =0.34
(Other than RBS) (Other than RBS)
Fitted Lognormal, =0.43 Fitted Lognormal, =0.43
0 0
0 1 2 3 4 0 1 2 3 4
(a) (b)
Figure 3-5 Cumulative distribution functions for reference cumulative plastic rotation, , for: (a) full
data sets; and (b) beam depths, d ≥ 21 in.
0.06
p (rad)
0.04
0.02
0
0 10 20 30 40
beam depth d (in)
Figure 3-6 Dependence of pre-capping plastic rotation, p, on beam depth,
d, for non-RBS connections, full data set.
Dependence on shear span to depth ratio, L/d. The plastic rotation capacity
for a given beam section is linearly proportional to the ratio between the
beam shear span, L (distance from plastic hinge location to point of
inflection) and depth, d. This proportionality is shown in Figure 3-7 for all
non-RBS data (beam depths ranging from 4 inches to 36 inches), but this
strong dependence on L/d does not hold true for beam depths larger than 21
inches.
Beams other than RBS : p versus L/d ratio
0.08 2
R = 0.278
0.06
p (rad)
0.04
0.02
0
0 5 10 15
L/d
Figure 3-7 Dependence of pre-capping plastic rotation, p, on shear span
to depth ratio, L/d, for non-RBS connections, full data set.
Dependence on Lb/ry. The ratio of Lb, defined as the distance from the
column face to the nearest lateral brace, and ry, the radius of gyration about
the y-axis of the beam, is associated with protection against premature
lateral-torsional buckling. Seismic codes require that this ratio be less than
2500/Fy. Data indicate that the pre-capping plastic rotation is somewhat, but
not greatly affected by Lb/ry, provided the ratio is close to or smaller than the
code-specified limit. Counterintuitively, providing lateral bracing close to
the RBS portion of a beam does not lead to a significant improvement in p.
A similar observation was made by Yu et al. (2000).
Dependence on h/tw. The depth to thickness ratio of the beam web, h/tw, is
important for all three modeling parameters. Figure 3-8 shows that all three
modeling parameters decrease with increasing h/tw ratios, for both RBS and
non-RBS connections.
0.06 0.06
p (rad)
p (rad)
0.04 0.04
0.02 0.02
0 0
0 20 40 60 80 0 20 40 60 80
h/tw h/tw
(a) Pre-capping plastic rotation, p; non-RBS (b) Pre-capping plastic rotation, p; RBS
Beams other than RBS (d>=21"): pc versus h/tw Beams with RBS (d>=21"): pc versus h/tw
0.6 2
0.6 2
R = 0.156 R = 0.220
0.5 0.5
0.4 0.4
pc (rad)
pc (rad)
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 20 40 60 80 0 20 40 60 80
h/tw h/tw
(c) Post-capping rotation, pc; non-RBS (d) Post-capping rotation, pc; RBS
Beams other than RBS (d>=21"): versus h/tw Beams with RBS (d>=21"): versus h/tw
4 2 4
R = 0.107 R2 = 0.481
3 3
2
2
1 1
0 0
0 20 40 60 80 0 20 40 60 80
h/tw h/tw
(e) Cumulative plastic rotation, ; non-RBS (f) Cumulative plastic rotation, ; RBS
Figure 3-8 Dependence of modeling parameters on h/tw, for beam depths d ≥ 21 in., and RBS
and non-RBS connections.
The coefficients and exponents are different for beams with RBS connections
and those with non-RBS connections, but the equations lead to relatively
similar predicted values, as shown in Table 3-1 and Table 3-2. The range of
experimental data covered by each parameter is listed along with the
definitions. Missing from the data set are results for heavy W14 sections and
heavy, deep beam sections. Values resulting from these equations, however,
were compared with data from a series of experiments on heavy W14
sections (Uang and Newell, 2007), and shown to provide conservative (low)
values of predicted modeling parameters. Until more tests on heavy sections
become available, the above equations represent the best available
information.
Table 3-2 Modeling Parameters for Various Beam Sizes (with RBS connections) Based on
Regression Equations with Assumed Beam Shear Span L=150 in., Lb/ry= 50, and
Expected Yield Strength, Fy=55 ksi
Section Size θp (rad) θpc (rad) Λ h/tw bf /2tf Lb /ry L/d d (cm)
W21x62 0.028 0.16 0.97 46.90 6.70 50.00 7.14 53
W21x147 0.033 0.27 2.15 26.10 5.43 50.00 6.79 56
W24x84 0.026 0.19 1.08 45.90 5.86 50.00 6.22 61
W24x207 0.030* 0.34* 2.71* 24.80 4.14 50.00 5.84 65
W27x94 0.022 0.16 0.91 49.50 6.70 50.00 5.58 68
W27x217 0.026* 0.29* 2.12* 28.70 4.70 50.00 5.28 72
W30x108 0.020 0.16 0.89 49.60 6.91 50.00 5.03 76
W30x235 0.023 0.25 1.78 32.20 5.03 50.00 4.79 80
W33x130 0.018 0.16 0.86 51.70 6.73 50.00 4.53 84
W33x241 0.020 0.22 1.46 35.90 5.68 50.00 4.39 87
W36x150 0.017 0.16 0.89 51.90 6.38 50.00 4.18 91
W36x210 0.019* 0.25* 1.53* 39.10 4.49 50.00 4.09 93
*Values slightly outside the range of experimental data
Flexural strength parameters for steel beams, including the yield strength,
maximum strength, and residual strength, can be quantified as follows:
Capping strength, Mc. Lignos and Krawinkler (2009) report a mean value of
the ratio of capping strength to effective yield strength, Mc/My, of 1.09 for
beams with RBS connections and 1.11 for beams with non-RBS connections.
A value of 1.1 for this ratio is recommended in both cases.
While they appear different, there are no evident conflicts between the data
presented here and the modeling parameters for fully restrained moment
connections presented in ASCE/SEI 41-06.
In ASCE/SEI 41-06, modeling parameters a and b are defined for a
model in which cyclic deterioration has already been taken into account.
Parameters a and b are intended to be used in a pushover analysis, and do
not provide information on how the post-capping negative tangent
stiffness range of component response should be modeled.
The values of pre-capping rotation, p, presented in Figure 3-3 appear to
be smaller than values for beams in flexure presented in Table 5-6 of
ASCE/SEI 41-06. The values in ASCE/SEI 41-06 are presented in terms
of multiples of y, which is a questionable practice because y is sensitive
to the beam shear span to depth (L/d) ratio. Moreover, the definition of
plastic rotation angle employed in ASCE/SEI 41-06 is different from the
definition of pre-capping rotation used here.
1.5c
pu
Modified backbone curve, Option 3
Mc
Initial backbone curve
0.8Mc
’p=0.7p
Ultimate rotation, Option 3
Figure 3-9 Procedure for obtaining the modified backbone curve for
modeling Option 3, and the ultimate rotation, u, for modeling
Option 4.
10
[Sa(T1)/g]/
8
6
4
2
0
0 1 2 3 4
Max. Strong Column Factor Over the Height, (2Mc/Mp,b)f,max
Figure 3-10 Strong column factor, R, required to avoid plastic hinging in
columns for a 9-story moment-resisting frame structure (Medina
and Krawinkler, 2003).
(a) (b)
Figure 3-11 Representative results from tests on W14x176 column sections subjected to an axial load
and cyclic bending moment: (a) moment versus story drift response for P/Py = 0.35; and (b)
peak moment versus story drift for P/Py = 0.75 (Newell and Uang, 2006).
Only a small number of tests are available for steel elements subjected to
combined axial load and inelastic deformations caused by cyclic bending
moments. None of the available data incorporate the variation in moment
gradient observed in nonlinear analytical studies, and no tests have been
performed on deep column sections. Analytical modeling, therefore, must be
based on a combination of incomplete column test data, principles of
mechanics, and extrapolation from beam test results.
The values listed for “Columns-flexure” in Table 5-6 of ASCE/SEI 41-06 are
believed to be unconservative, and should be re-evaluated. Also, they should
be expressed in terms of plastic rotation angle, p, rather than multiples of
yield rotation, y, since the moment gradient is expected to change
significantly during inelastic response.
Local acceptance criteria for steel beam and column components are
provided for service level and Maximum Considered Earthquake (MCE)
level evaluations. Additional local component acceptance criteria, and global
acceptance criteria for the overall structural system, are provided in Seismic
Design Guidelines for Tall Buildings (PEER, 2010).
Mathematical models for the behavior of the panel zone in terms of shear
force-shear distortion relationships have been proposed by many researchers,
including Krawinkler (1978), Tsai and Popov (1988), Kim and Engelhardt
(1995), and Jin and El-Tawil (2005), based on either experimental
observations or finite element modeling. The models differ in their
representation of inelastic behavior, but agree well in their representation of
the elastic shear stiffness, Ke, and the yield strength in shear, Vy.
In frame analysis programs that utilize line elements, panel zone behavior
can be modeled by creating a panel zone with rigid boundaries as illustrated
in Figure 3-13. The model requires the use of 8 rigid elements per panel
zone that are connected with hinges at the four corners. These 8 rigid
elements create an assembly that deforms into a parallelogram. The strength
and stiffness properties of the panel zone can be modeled by adding one (or
two) rotational springs to one of the four panel zone corners, or by adding
Column
2 Rotational
Springs
Rotational
Spring
Beam
db
Rigid
Element
dc
Figure 3-13 Analytical model for panel zone (Gupta and Krawinkler, 1999).
Under lateral deformation, steel panel zones are subjected to large shear
forces. If the panel zone is relatively thin, shear yielding will propagate from
the center towards the boundaries, followed by inelastic bending of the
adjacent column flanges. This behavior is characterized by stable inelastic
cycles of the type shown in Figure 3-14.
Figure 3-14 Cyclic shear behavior of weak panel zone (Krawinkler, 1978).
If the panel zone is relatively strong, and only limited inelastic behavior is
expected, then a simple bilinear hysteretic model should be sufficient. If the
panel zone is expected to contribute significantly to the inelastic story drift,
then a trilinear model is recommended to account for an increase in strength
that occurs after initial yielding, which cannot be adequately captured in a
bilinear model.
Fy Fy
Vy Aeff (0.95d c t p ) 0.55Fy d c t p (3-7)
3 3
where Vy is the panel zone shear yield strength, Fy is the yield strength of the
material, Aeff is the effective shear area, dc is the depth of the column, and tp is
the thickness of the web including any doubler plates. The corresponding
yield distortion, y, is given as:
Fy
y (3-8)
3 G
The elastic stiffness, Ke, of the panel zone can then be written as:
Vy
Ke 0.95d c t p G (3-9)
y
3K p 3bc t cf 2
V p V y 1 0.55Fy d c t p 1 (3-10)
Ke db dct p
where Kp is the post-yield stiffness, bc is the width of the column flange, and
tcf is the thickness of the column flange. This strength is assumed to be
attained at a value of 4y. Beyond 4y, an appropriate value of strain-
hardening can be assumed to fully define the trilinear shear force and shear
distortion relationship of the panel zones.
V
Vp Ke
Kp
Vy
Ke
y p
Figure 3-15 Trilinear shear force and shear distortion relationship for panel
zone (Gupta and Krawinkler, 1999).
The shear force demand on the panel zone, V, can be estimated using the
following equation:
M
V Vcol (3-11)
db
where M = Mbl + Mbr, which is the net beam moment transferred to the
column, and Vcol represents the average of the shears in the column above
and below the connection, as shown in Figure 3-16.
If rotational springs are used in one corner, the total spring stiffness is given
as db(V/. The use of two bilinear springs to model panel zone trilinear
behavior is illustrated in Figure 3-17. A representative example of the panel
zone dynamic response obtained with this model is shown in Figure 3-18.
Beam Vcol,b
Mcb
dc
Figure 3-16 Moment and shear forces at a connection due to lateral loads.
Ke
Vp
Kp
Vy K p1 =0
Ke K p2 = K e
K e1 = K e - K p
K e2 = K p
y 4 y
Figure 3-17 Use of two springs to model trilinear behavior (Gupta and
Krawinkler, 1999).
400
-400
-800
-0.02 -0.01 0 0.01 0.02 0.03 0.04
Panel Zone Distortion (radians)
Figure 3-18 Shear force-distortion response for a typical panel zone (Gupta
and Krawinkler, 1999).
The model illustrated here has been calibrated for cases in which the column
flange thickness is less than 10% of the column depth. Recent research, such
as Jin and El-Tawil (2005), has shown that this model overestimates the
Local acceptance criteria for steel panel zones are provided for service level
and MCE level evaluations. Additional local and global acceptance criteria
are provided in Seismic Design Guidelines for Tall Buildings (PEER, 2010).
At the service level, the shear force demand in the panel zone should be less
than 1.5 times the yield strength, Vy, if elastic analysis is used. This value is
judgmental, and is based on the following two assumptions: (1) no remedial
action needs to be taken; and (2) the effect of panel zone yielding on the
deformation capacity of the beam-column connection is accounted for in
acceptance criteria at the MCE response level.
For frames in which panel zone shear distortion does not contribute to the
incident of fractures at the beam-column connection, a shear distortion angle
of 0.08 radians, presently accepted for link elements in eccentrically braced
frames in AISC 341 (AISC, 2005b), should be used.
If panel zone shear distortion causes kinking at the panel zone corners that
contributes to fractures at the beam-column connection, then the shear
distortion angle should be limited to 0.02 radians, unless a larger (or smaller)
value is justified based on experimental evidence.
(a)
1.5
M p pc
1
Mc
Moment (M/My)
0.5
0 My
-0.5
Mr
-1 Ke
-1.5
-8 -6 -4 -2 0 2 4 6 8
Chord Rotation (radian x 10-2)
(b) (c)
Figure 3-19 Reinforced concrete flexural member: (a) idealized flexural
element; (b) monotonic backbone curve and hysteretic
response; and (c) monotonic and modified backbone curves.
Panel zone Column Spring
Beam Spring
Joint Panel Spring
Beam‐end zone
Column‐end zone
Figure 3-20 Idealization of reinforced concrete beam-column joint.
In this idealization, each of the four springs connecting the joint to the
adjacent beam or column is common to the joint and the connected element.
As such, these springs are calibrated to model both the inelastic deformations
in the member plastic hinge and the bond-slip/yield penetration into the joint,
and are the same as the springs at each end of the flexural member in Figure
3-19. The spring at the center of the joint is calibrated to model joint panel
deformations due to the large shear force transfer through the joint. Clear
distinction and accounting of deformations associated with the flexural
response of the member, anchorage into the joint, and joint panel shear are
important when calibrating flexural hinges for reinforced concrete frame
components.
Since the design of new tall buildings is expected to conform to current code
and reference standard seismic design and detailing requirements, such as
ACI 318 Building Code Requirements for Structural Concrete (ACI, 2008),
recommended modeling and acceptance criteria are based on the following
behavioral assumptions:
Member shear design: It is assumed that shear capacity design
provisions are in effect, which will prevent premature shear failures in
beams and columns. Thus, while shear strength demand to capacity
ratios should be checked, the inelastic hinge calibration is for
components that are dominated by flexural effects, considering the
interaction of axial load and moment.
Joint panel design: It is assumed that the joint panel will be designed to
resist shear and bar anchorage forces associated with flexural hinging of
the connected members, which will preclude joint shear failure, limit
bond slip, and prevent bar pullout. The joint shear strength demand to
capacity ratios should be checked, but large inelastic panel deformations
should not be permitted.
Longitudinal reinforcing bars: It is assumed that the longitudinal
reinforcing bar splices in beams and columns will be designed and
detailed to prevent splice failure that would otherwise limit flexural
hinging in the members and cause sudden strength degradation.
100
Fy
80
Force (kN)
60
40
0.4Fy
20
θstf_40 θy
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Chord Rotation (rad)
Haselton et al. (2008) proposed the following equations for median estimates
of Ky and Kstf, based on calibration to tests of reinforced concrete columns:
EI y P Ls EI y
Ky: 0.07 0.59 0.07 where 0.2 0.6 (3-12)
EIg Ag fc H EI g
EI stf P Ls EI stf
Kstf: 0.02 0.98 0.09 where 0.35 0.8 (3-13)
EI g Ag fc H EI g
In these equations, EIg is the flexural stiffness of the gross section, P is the
member axial (compression) load, Ag is the gross column area, f c is the
concrete compressive stress, Ls is the shear span from the point of maximum
moment to the inflection point (typically one-half of the member length), and
H is the member depth. For median value estimates, material properties used
to calculate the stiffness parameters should be based on expected values.
Variations in stiffness tend to follow a lognormal distribution with a variation
of ln = 0.28 for Ky and ln = 0.33 for Kstf. Note that these stiffness values
Elwood et al. (2007) recommended effective stiffness values that have been
adopted in ASCE/SEI 41-06 Supplement No. 1. These values range from
0.3EIg to 0.7EIg, which are also lower than values that are commonly
assumed. Figure 3-22 shows a comparison of the values contained in
ASCE/SEI 41-06 Supplement No. 1 (designated ASCE 41S in the figure) to
the values obtained using Equation 3-12 and Equation 3-13.
Kstf_40
ASCE 41S
Ky
Overall, the values given by the equations (Ky and Kstf_40) tend to bracket the
values in ASCE/SEI 41-06 Supplement No. 1. While based on similar data
and criteria, differences are likely due to variations in how the underlying
data were processed to recover EIeff, and differences in the intended statistical
adjustments. While the values given by Equations 3-12 and 3-13 are
There are relatively few data available on reinforced concrete beams that are
integral with the floor slab or that have post-tensioned reinforcing. In the
absence of other data or information, the values for column stiffness at zero
axial load can be used for beams without post-tensioning, where the EIg is
adjusted to account for the presence of a slab and reverse curvature bending.
In reverse curvature bending, EIg may be taken as the average of the gross
stiffness for positive bending (based on an effective slab width equal to one-
eighth the beam span on each side of the beam) and the gross stiffness of the
beam section alone for negative bending. For beams with post-tensioned
slabs, stiffness should be increased based on the axial load ratio induced by
the post-tensioning over the effective slab width.
The equation was calibrated to match the median response from column
tests, with a reported dispersion (standard deviation of the logarithm of the
data) of ln = 0.54. To the extent that the reinforcement in beams is similar
to the columns on which these data are based, Equation 3-14 can be applied
to beams.
' fy
max 0.01,
f c
p ( non symmetric ) p ( symmetric ) (3-15)
fy
max 0.01,
f c
This equation was calibrated to the median response from column tests, with
a reported dispersion of ln = 0.72. The larger dispersion for post-capping
rotation reflects both the larger inherent uncertainty in degrading behavior,
and a relative lack of available data. The upper bound of pc< 0.10 is a
conservative assumption based on limited availability of data for elements
with shallow post-capping slopes. Since this ultimate rotation is fairly large,
and data to quantify the response at large deformations are lacking, the
residual strength of the hinge, Mr, should be conservatively neglected, and
taken as zero (or near zero).
Equation 3-16 is based on data from square and rectangular columns with
symmetric reinforcement. Presumably, the equation can be similarly
adjusted for cross sections with non-symmetric reinforcement, and applied to
beams, using the modifier in Equation 3-15.
For columns with low axial load ( 0.1 f cAg ), the pre-capping rotation ranges
from p = 0.031 radians for a column with minimal confinement, up to p =
0.077 radians for a column with heavy confinement. For columns with
higher axial loads ( 0.6 f cAg ) above the balance point, the pre-capping
rotation is reduced to less than half of the corresponding values at lower axial
loads, ranging from p = 0.012 to 0.031 radians. In both cases, bond slip
(incorporated through the sl parameter) accounts for about one-third of the
total pre-capping rotation.
Similar to the trends for pre-capping rotation, post-capping rotation drops off
dramatically for axial loads above the balance point and low confinement
ratios. Values for post-capping rotation range from pc = 0.10 for a column
with low axial load and high confinement, down to pc = 0.009 for a column
with high axial load and low confinement.
(30)(0.03) v (3-17)
where is the axial load ratio. This equation was calibrated to the median
response from column tests, with a reported dispersion of ln =0.60. For a
typical column with seismic detailing, typical values of the parameter are
Capping strength, Mc. While the capping strength, Mc, can, in theory, be
determined through analysis, its calculation is complicated by assumptions
regarding steel strain-hardening, concrete stress-strain behavior, and other
factors. For the initial backbone curve, Mc, should incorporate the effects of
cyclic hardening of steel. Regression analysis by Haselton et al. (2008)
showed that the capping strength can be estimated from My by assuming a
constant ratio of Mc/My = 1.13. The additional variability introduced by this
ratio is σln=0.10, which, when combined with the variability in My, results in
a total variability on Mc that is equal to σln=0.37.
Residual strength, Mr. Since values of ultimate plastic rotation, pc, are fairly
large, and data to quantify the response at large deformations are lacking, the
residual strength, Mr, should be conservatively neglected, and taken as zero
(or near zero).
Axial Load (P/Agf'c)
0.6 0.7Theta‐p [p"=0.002]
0.7Theta‐p [p"=0.006]
0.5
ASCE 41 ‐ beam
0.4
0.3
0.2
0.1
0
0 0.01 0.02 0.03 0.04 0.05
Plastic Rotation (radians)
(a)
1
ASCE 41 Col‐i [p"=0.002]
0.9 ASCE 41 Col‐ii[p"=0.002, high V]
0.8 ASCE41 Col‐i[p"=0.006]
ASCE41 Col‐ii [p"=0.006, high V]
0.7
Axial Load (P/Agf'c)
0.7Theta‐p+0.5Theta‐pc [p"=0.002]
0.6
0.7Theta‐p+0.5Theta‐pc [p"=0.006]
0.5 ASCE 41 ‐ beam
0.4
0.3
0.2
0.1
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Plastic Rotation (radians)
(b)
Figure 3-24 Comparison of plastic rotation parameters for modeling Option
3 versus ASCE/SEI 41-06 Supplement No. 1 for: (a) pre-capping
rotation capacity; and (b) post-capping rotation capacity.
In Option 4, the ultimate rotation is taken at the point of 20% strength loss on
the descending branch of the modified backbone curve. Using the approach
illustrated for steel beams in Figure 3-9, the resulting ultimate rotation limit
would be 0.7p + 0.1pc. In Option 3, a rotation limit of 1.5c, or
alternatively, a corresponding plastic rotation limit of 1.5p, is used.
Plots show reasonably good agreement between the values for Option 4 and
ASCE/SEI 41-06 Collapse Prevention limits for primary components, but
values for Option 3 appear unconservative relative to ASCE/SEI 41-06
Collapse Prevention limits for secondary components. As noted previously,
in cases with high shear demands, the discrepancies are larger, due, in part, to
a conservative penalty that ASCE/SEI 41-06 places on components with high
shear force demands.
0.9 ASCE 41 Col‐i [p"=0.002]
ASCE 41 Col‐ii[p"=0.002, high V]
0.8
ASCE41 Col‐i[p"=0.006]
0.7 ASCE41 Col‐ii [p"=0.006, high V]
Axial Load (P/Agf'c)
0.6 Theta‐u [p"=0.002]
Theta‐u [p"=0.006]
0.5
ASCE 41 ‐ beam
0.4
0.3
0.2
0.1
0
0 0.01 0.02 0.03 0.04 0.05
Plastic Rotation (radians)
(a)
1
ASCE 41 Col‐i [p"=0.002]
0.9
ASCE 41 Col‐ii[p"=0.002, high V]
0.8
ASCE41 Col‐i[p"=0.006]
0.7 ASCE41 Col‐ii [p"=0.006, high V]
Axial Load (P/Agf'c)
0.6 1.5theta‐c [p"=0.002]
1.5theta‐c [p"=0.006]
0.5
ASCE 41 ‐ beam
0.4
0.3
0.2
0.1
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Plastic Rotation (radians)
(b)
Figure 3-25 Comparison of ultimate plastic rotation versus ASCE/SEI 41-06
Supplement No. 1 acceptance criteria at the Collapse Prevention
performance level for: (a) Option 4; and (b) Option 3.
The finite joint size and shear panel deformations can generally be included
explicitly, through a joint panel model as shown in Figure 3-20, or more
approximately by defining effective rigid end offsets for elements framing
into the joint. When modeled explicitly, the joint stiffness can be calculated
using modified compression field theory (Mitra and Lowes, 2007) or by
calibrating the effective initial stiffness to test data.
Figure 3-26 Recommended rigid end zone offsets for reinforced concrete beam column joints
based on relative column and beam strengths (Elwood et al., 2007).
M
Elastic column at
Wall wall centroid
Rigid
Links (c) Rigid-plastic hinge
Beams
Shear
M
spring
Hinges
Concrete Fibers 4
3
2
Steel Fibers 1 Element
Use of a fiber model, with defined uniaxial material relations, implies several
important differences relative to the use of an equivalent beam-column
model. In an equivalent beam-column model, the elastic flexural stiffness is
specified (e.g., EcIeff = 0.5EcIg), whereas in a fiber model, the flexural
stiffness is derived from the specified material relations, and varies
depending on the magnitude of axial load. Also, in a beam-column model,
the flexural strength is defined using simplified concrete theory (i.e.,
prescribed strains in the extreme steel or concrete fibers, linear concrete
compressive behavior at yield moment, or Whitney Stress Block at nominal
moment), whereas in a fiber model, the section strength depends on the
prescribed material relations.
Modeling of core wall systems using detailed finite element models with
concrete elements (e.g., brick elements) and discrete modeling of
reinforcement is possible with currently available commercial software.
Although such models are available, they should be used with caution, as it is
often difficult to determine if the resulting load versus deformation behavior
of the components is within reasonable bounds. Given this difficulty, use of
detailed nonlinear finite element models is not emphasized.
800
400
Flexural Analysis
Test
Analysis
0
0 0.4 0.8 1.2 1.6 2
Lateral Displacement (cm)
Figure 4-4 Coupled model and results for a low-aspect ratio wall (Massone,
2006).
There are few studies focusing on the force versus deformation response of
walls governed by shear behavior. Deformations corresponding to the onset
of yield and shear strength degradation are based on limited test data, such as
Hidalgo et al. (2002), Hirosawa (1975), and Massone (2006). Figure 4-5a
shows the shear force-deformation relation (backbone curve) provided in
FEMA 356, Prestandard and Commentary for the Seismic Rehabilitation of
Buildings (FEMA, 2000d). Figure 4-5b shows an improved relation that is
provided in ASCE/SEI 41-06 Supplement No. 1 (ASCE, 2007b), which
allows the backbone curve to be modified to include a pre-cracked stiffness
and strength, followed by post-cracked stiffness up to the nominal (yield)
strength.
1.0 Vn
V Vcr
Vn
0.4Ec Vr
0.2 0.4Ec c
c
The nominal shear strength of walls is typically defined using Equation 4-1,
taken from ACI 318-08:
Vn Acv c f c t f y (4-1)
where c = 3.0 for a height-to-length ratio, hw/lw < 1.5, c = 2.0 for hw/lw
2.0, and varies linearly for 1.5 < hw/lw < 2.0. In this equation, is 0.75 for
lightweight concrete and 1.0 for normal weight concrete, Acv represents the
cross-sectional web area of a wall, f c is the compressive strength of concrete,
t is transverse reinforcement ratio, and fy, is the yield strength of transverse
reinforcement. The variation of c for hw/lw values between 1.5 and 2.0
accounts for the observed strength increase for low-aspect ratio walls. An
upper limit on nominal shear strength is set at Vn Acv 10 f c for a single
wall, which is the same limit used for beams (ASCE-ACI, 1973), and
Limiting the wall shear stress to Vn Acv 6 f c has been suggested (Wang
et al., 1975; Aktan et al., 1985) to ensure ductile response for reverse cyclic
loading, and to avoid sliding shear failures, unless diagonal shear
reinforcement is present (Paulay, 1980). Subsequent test results reported by
Paulay et al. (1982) for walls controlled by flexural yielding indicated that
ductile response, with displacement ductility ratios exceeding four, could be
achieved for walls with a maximum shear stress of approximately
Vn Acv 8 f c , provided failure modes associated with diagonal tension,
diagonal compression, and sliding shear were prevented. Test results also
indicated that flanged walls, because of the reduced flexural compression
depth, were more susceptible to shear strength degradation associated with
sliding shear.
In contrast to shear strength of columns, the ACI 318-08 equation for shear
strength of walls does not consider the effect of axial load. Orakcal et al.
(2009) reported that wall shear strength is sensitive to axial load, with Vtest /Vn
values of approximately 1.5 for walls tested with an axial load of
Pu / Ag f c 0.05 , and 1.75 for walls with an axial load of Pu / Ag f c 0.10 .
Relatively few wall tests with reported shear failures have been conducted
that include axial load, so insufficient information exists to systematically
assess the impact of axial load on shear strength.
where Pu is the factored axial load, and Aweb is the cross-sectional area of the
web, and other parameters are as defined in Equation 4-1.
Oesterle et al. (1984) used a truss analogy to determine the shear stress
associated with web crushing of barbell and I-shaped wall cross-sections as:
Nu
vn , wc 0.14 f c 0.18 f c (4-3)
2lw h
Test data for walls with concrete strength exceeding 10 ksi are summarized
by Kabeyasawa and Hiraishi (1998) and Farvashany et al. (2008).
Kabeyasawa and Hiraishi (1998) reported on tests with concrete strengths
ranging between 10 ksi and 15 ksi, and shear-span-to-depth ratios, Mu/VuIw ,
between 0.6 and 2.0. Wallace (1998) evaluated these data with respect to
ACI 318-95 requirements (which are essentially equivalent to ACI 318-08).
The median ratio of Vtest /Vn was 1.38, with a standard deviation of 0.34,
indicating that ACI 318 requirements provided a lower-bound estimate of
tested wall shear strengths. For ratios of nfy / f c > 0.10, ACI 318 tended to
overestimate tested shear strengths, indicating that the contribution of web
reinforcement to shear strength was overestimated. This is consistent with
results reported by Wood (1990) for concrete compressive strengths ranging
from 2.0 ksi to 6.0 ksi.
Models that reduce shear strength with increasing ductility demand have
been provided in references for design of columns, such as Sezen and
Moehle (2004), Zhu et al. (2007), and ASCE/SEI 41-06. An analogous shear
strength capacity relation for walls is:
Vn k Acv c f c t f y (4-4)
In the case of walls, relatively sparse data exist for judging whether shear
strength should be degraded with increasing ductility demand. Results from
two small scale tests reported by Corley et al. (1981) show that deformation
capacity is impacted by the level of shear stress, i.e., wall shear strength
degrades with increasing ductility demand. Oesterle et al. (1984) suggest
0.4
Shear Failure
Flexure/Shear Failure
Flexure Failure
Shear Strength
0
0 2 4 6
Ductility (max/y)
A specific recommendation for shear walls is not given here, as this issue
requires further study. However, for performance-based design of core wall
systems in tall buildings, it would appear prudent to consider some reduction
in shear strength with increasing ductility demand.
where is Poisson’s ratio, and Acv is the cross-sectional area of the web.
Based on the assumption that Poisson’s ratio for uncracked concrete is
approximately 0.2, the effective shear stiffness defined in ASCE/SEI 41-06 is
GcA = 0.4EcAw.
From Mohr’s circle, the shear strain at yield is twice the principal strain
(approximately 0.002 for Grade 60 reinforcement); therefore, the effective
shear stiffness at yield is approximately Gc/20 for walls with a shear strength
of 5 f c Acv . For typical walls in new construction, the effective shear
stiffness at yield doubles to approximately Gc/10 at a shear strength of about
10 fc Acv .
Sozen and Moehle (1993) studied load versus deformation behavior of low-
rise walls and reported that a relatively simple model could be used to
reasonably capture the measured load versus deformation response prior to
strength degradation. The proposed model consists of three components to
account for flexure, shear, and slip. For flexure, the force versus deformation
behavior is defined for two points, cracking and yielding, along with an
assumed post-yield stiffness equal to 3% to 15% of the yield stiffness. The
moment capacities at cracking and yielding, and the associated deformations,
are calculated using straightforward methods, such as those based on the
results of a moment-curvature analysis.
For shear, two points, cracking and ultimate, are used to define the load
versus deformation behavior. The cracking point is defined by:
fo vc
vc 4 f c 1 c k (4-6)
4 f c Gc
where vcu is the ultimate shear stress in psi, cu is the shear strain at shear
capacity, w is the wall web reinforcement ratio (lesser of the two
directions), n is the modular ratio, Es /Ec , and f y is the reinforcement yield
stress. To convert from stress to force, the stresses at each point are
multiplied by the area of the web.
The effective shear stiffness of low-rise walls was also examined by Elwood
et al. (2007), focusing on lightly-reinforced wall segments. Figure 4-7 shows
that the use of 0.4Ec is reasonable for modeling uncracked shear stiffness of
lightly reinforced wall piers, provided that the flexural deformations are
modeled independently (e.g., with a fiber model).
Drift (%) Drift (%)
-0.02 -0.01 0.00 0.01 0.02 -0.020 -0.010 0.000 0.010 0.020
150
0 0
-50
-50
-100
-100
-150
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00
Lateral Displacement (in.) Lateral Displacement (in.)
Given the influence of axial load and web reinforcement on shear strength,
and the effects of cracking, selection of a single effective shear stiffness is
problematic for analysis of shear wall systems. The impact of potential
uncertainty in the effective shear stiffness of the wall should be considered to
obtain bounds on important response quantities.
1.5
Analysis: Ieff=0.4Ig
Displacement (in.)
Measured
-1.5
0 10 20 30
Time (sec)
(a) 10-story walls (Yan and Wallace,1993)
0
40 45 50 55 60 65 70 75 80
-2
-4
-6 UCSD Test
ETABS
-8
Time [s]
(b) 7-story walls (Panagiotou and Restrepo, 2007)
Figure 4-8 Roof displacement response correlation studies for: (a) 10-story
walls; and (b) 7-story walls.
A value of 0.4EcIg was used with axial loads on the order of P 0.10 Ag f c ,
and a value of 0.5EcIg was used as a reasonable approximation for walls with
modest axial load. However, displacement response history comparisons at
the top of a 7-story wall tested by Panagiotou and Restrepo (2007) indicate
that 0.2EcIg produced good agreement between tested and modeled results
(Figure 4-8b). These disparate results indicate that appropriate effective
stiffness values may vary depending on the specific characteristics of the
wall.
0.7
0.6
0.5
Ie/ Ig
0.4
0.3
Lower Bound
Upper Bound
0.2
0.1
0
0 0.05 0.1 0.15 0.2
P/f'c Ag
(a) Effective stiffness versus axial load ratio (b) Effective stiffness versus displacement ratio
Figure 4-9 Upper-bound and lower-bound wall flexural stiffness versus: (a) axial load ratio; and
(b) displacement ratio (Adebar et al., 2007).
Mn2
Mn1
sy c 0.003
y =
lw lw
Mn
EI effective
y
y
Figure 4-10 Impact of wall flexural strength on effective stiffness.
The test presented in Figure 4-8b was conducted for a relatively low axial
load P 0.05 Ag f c , and a primary objective of that study was to
demonstrate that satisfactory lateral-load behavior could be achieved using
approximately one-half the longitudinal reinforcement typically required in
current codes. The lower value of effective stiffness observed in that test
might, therefore, be an artifact of the test parameters.
P = 0.07Agf'c
0
0
-40
-20
-80
Flat@Mn(c=0.003)=77.0
vu,max = 5.4f'c psi
-40 -120
-4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0
Top Displacement (in.) Top Displacement (in.)
For lower levels of axial load, use of Equation 4-8b for the lower-bound
effective stiffness is recommended to account for variation in axial load.
Results from Adebar et al. (2007) suggest that higher axial stress levels,
which would be expected to reduce cracking, are likely to yield higher
effective stiffness values. At this time, however, there are insufficient test
data available to assess the appropriateness of using higher effective linear
stiffness values for higher levels of axial load.
Moment (kN-m)
4000
Moment (k-in)
400 120000
3000
300
80000
2000
200 IN, at g.l.
Analytical - BIAX FOS, 2 ft above g.l.
FOS, 2 ft below g.l.
1000 Measured - LVDTs 100 EX, 1 ft below g.l.
40000
RW3-O Section analysis: average rebar
0 0
0
0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0 0.0004 0.0008 0.0012
Curvature (/in) Curvature, (1/in)
In fiber element models, effective stiffness values are not used since load
versus deformation response of a fiber model depends on the uniaxial
material stress-strain relations specified for the concrete and steel fibers,
level of axial load, and current condition of the element (considering
hysteretic response in nonlinear response history analysis). Example stress-
strain relations used to capture the uniaxial cyclic behavior of concrete and
steel reinforcement are shown in Figure 4-13.
1.5
2
/y
1 R0
0.5
Normalized Stress,
R(2)
0
R(1)
a1
-0.5
R = R0 - a +
2
-1 ( m - 0 )
=
1 yy
-1.5
Stress, fc
O Tension
Not to scale
(0+ t, ft)
Strain, c Strain, c
(b) Concrete (Yassin, 1994) (c) Concrete (Orakcal and Wallace, 2004)
Common concrete material models include Yassin (1994) and Orakcal and
Wallace (2004). The envelope of the concrete model in Figure 4-13b is
based on the model from Scott et al. (1982), which considers both
unconfined and confined concrete behavior, with the unloading and reloading
rules described by Yassin (1994). Additional models are available, including
envelope models proposed by Mander et al. (1988), Chang and Mander
(1994), and Saatcioglu and Razvi (1992), and Razvi and Saatcioglu (1999).
80 80
40 40
0 0
-40 -40
-80 -80
-120 -120
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
Steel Strain (in/in) Steel Strain (in/in)
(a) Reinforcing steel model (SH) (b) Reinforcing steel model (EPP)
8
6
Concrete Stress (ksi)
0
0 0.002 0.004 0.006 0.008 0.01
Concrete Strain (in/in)
An elevation of the fiber model is shown in Figure 4-15a, and the resulting
strain gradient for the rectangular wall is shown in Figure 4-15b. Results
show good agreement between the simple model (Perform 3D), complex
model (MVLE), and test results at smaller drift ratios (e.g., on the order of
0.5%). At larger drift ratios, where differences in the unloading and
reloading rules have more impact, somewhat more variation is observed
between the results for different material models and the test results. The
most significant discrepancies occur in the predicted concrete compressive
strains, but the results obtained using both sets of material models are
reasonably accurate.
0.01
0.5 % Drift
12 in.
-0.01
12 in.
12 in.
0 10 20 30 40 50
Distance Along Web (in)
20
0
10
-20
0
-40
-10
-60
-20
-30 -80
Figure 4-16 Behavior of a rectangular wall section subjected to constant axial load
and reverse cyclic loading.
Results from a fiber model using fairly sophisticated material models for the
concrete and reinforcing steel are presented in Figure 4-17. The model is
capable of simulating the top flexural displacement measured in the tests, but
200
Pax 0.07A gf 'c
Test
150 Analysis
Plat , top
Lateral Load, Plat (kN)
100
RW2
50
-50
500
Pax (kN)
-100 400
300
200
-150 100
0
-200
Figure 4-17 Comparison of model and test results for a rectangular wall
section (Orakcal and Wallace, 2006).
The lateral load versus top displacement response using two different
concrete material models (see Figure 4-13), are compared in Figure 4-18.
100
Lateral Load (kN)
-100
Concrete Constitutive Models:
Chang and Mander (1994)
Yassin (1994)
-200
Figure 4-19 shows a comparison of the average strain measured at the base of
the wall with results obtained from an analytical model. Figure 4-20 shows
curvature profiles (strain distribution over the cross-section) for three levels
of drift.
0.035
0.004
Concrete Strain Gage
0.03 LVDT (Linear Interpol'n)
0.002
Analysis
0.025
0
Concrete Strain
0.02
-0.002
0.015
-0.004
100 150 200 250 300
0.01
0.005 RW2
0
0
0.04 Test
Concrete Strain
Analysis -0.0025
0.03
-0.005
0.02 -0.0075
-0.01
0.01
0
0.5%
-0.01 1.0%
2.0%
-0.02
At 1.5% and 2.0% drift, substantial concrete spalling was observed at the
wall boundary opposite the flange. This is due to the larger moment capacity
in the negative loading direction (flange in tension), due to the contribution
of the longitudinal reinforcement in the flange. In contrast, concrete
compressive strains were relatively low, and no spalling was observed in the
positive loading direction (flange in compression), even for drift ratios
approaching 3.0%.
Results from a fiber model of the T-shaped wall section are presented in
Figure 4-22. The model was capable of reasonably simulating the top
flexural displacement measured in the tests, but somewhat overpredicted the
strength of the wall when the flange was in tension. The likely reason for
0
T
-100
-200 C
Pax (kN)
750
500
-300
250
-400 0
Figure 4-22 Comparison of model and test results for a T-shaped wall
(Orakcal and Wallace, 2006).
Figure 4-23 shows the distribution of concrete strains in the flange for
tension loading (negative displacements) and compression loading (positive
displacements). With the flange in compression, the strain distribution was
essentially uniform, indicating that the effective flange width in compression
did not vary significantly with drift ratio.
0.025
C
Test
0.015 Analysis T
0.5%
0.01
1.0%
2.0%
2.5% T
0.005
Positive Displacements C
0
-0.005
TW2
0.025
Test Negative Displacements
Flange Steel Strain
0.02 Analysis
0.015
1.0%
2.0%
0.01
2.5%
0.005
y
0
Positive Displacements
-0.005
1.5%
0.025 Positive Displacements -0.00125
0.02 T
-0.0025
0.015
C
0.01
0.005
0
-0.005
0.01 Negative Displacements
Concrete Strain
0.005 C
T
0
-0.005
-0.01
It is well known that results obtained using finite element analyses can be
sensitive to the mesh (or element size) used to define the structure or
structural component. It is somewhat natural to assume that the use of more
elements, especially within a yielding region, is advantageous. However,
predicted material strains in reinforced concrete walls can vary substantially
with differences in element height or length, depending on the material and
model parameters used.
In the sections that follow, model and test results for lateral load versus top
displacement response and element (local) strains are compared for a
rectangular wall section. Additional information regarding the sensitivity of
results to variations in wall model and material parameters are available in
Orakcal et al. (2004) and Orakcal and Wallace (2006).
20 20
Base Shear (K)
-20 -20
-40 -40
-0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02
Drift Drift
(a) (b)
Figure 4-26 Influence of reinforcing steel stress-strain relation on force-deformation
response for: (a) elastic-perfectly-plastic; and (b) strain hardening behavior.
(a) (b)
Figure 4-27 Influence of mesh size on force-deformation response for: (a) 91 elements;
and (b) six elements.
0.1
0.0175
0.06
0.01
0.01
0.08
0.0025 0.0025
0.04
-0.005 -0.005
0.06
-0.0125 2.0 % Drift -0.0125
Concrete Strain
Concrete Strain
St. Hard.
St. Hard.
EPP
0.04 EPP 0.02
2.0 % Drift Test
Test
1.5 % Drift
0.02
1.0 % Drift
0
0 0.5 % Drift
0.5 % Drift
-0.02
(a) -0.02
(b)
0 10 20 30 40 50 0 10 20 30 40 50
Distance Along Web (in) Distance Along Web (in)
0.04 0.01
(a) 0.0075
(b)
2.0 % Drift
0.03
0.005
0.0025
0.02
Concrete Strain
Concrete Strain
0.01 -0.0025
0.5 % Drift
-0.005
0
-0.0075
St. Hard. St. Hard.
EPP EPP
-0.01
-0.01 Test Test
-0.0125
0 10 20 30 40 50 36 40 44 48
Distance Along Web (in) Distance Along Web (in)
(c) 6 elements (d) 6 elements - detail
6. Due to the sensitivity of shear stiffness with respect to cracking and axial
load, uncertainty in the effective shear stiffness should be considered in
A consistent finding from coupling beam studies is that the use of diagonal
reinforcement improves the cyclic performance of beams with clear span-to-
depth ratios ratios less than about four. For ratios greater than four, use of
diagonal reinforcement is not practical given the shallow angle of the bars.
Information on coupling beam effective stiffness, detailing, force-
deformation behavior, and modeling is discussed in the sections that follow.
where h is the total depth of the coupling beam, ln is the clear span, and Ig is
the gross concrete cross-section moment of inertia. Coefficients A, B, and C
are provided in Table 4-1, based on the type of longitudinal reinforcement
(diagonal or conventional) and on the anticipated ductility demand.
NZS 3101 values for effective moment of inertia are intended for use with
linear analysis, and are secant approximations at the given level of ductility.
Values of the ratio Ie/Ig for a range of ductility demands () and clear span-
to-depth ratios are shown in Figure 4-29. For low ductility demands
implying modest yielding (=1.25), NZS 3101 values are close to the
ASCE/SEI 41-06 value of 0.5Ig. For ductility demands of =3.0 and =4.5,
NZS 3101 values are similar to the ACI 318-08 value (0.35Ig) and the
ASCE/SEI 41-06 Supplement No. 1 value (0.3Ig) at clear span-to-depth ratios
larger than 2.0.
Where a linear analysis is used for service level assessments, use of EcIeff =
0.3EcIg appears appropriate. If a linear analysis is used for a design level
0.5
0.4
Ieffective / Igross
0.3
0.2
=1.25
=3.0
0.1 =4.5
=6.0
0
1 2 3 4
Beam Clear Length / Beam Height
Figure 4-29 Coupling beam effective flexural stiffness ratios (based on NZS
3101).
Prior code provisions for diagonally reinforced coupling beams (e.g., ACI
318-05) resulted in designs with substantial rebar congestion, especially at
the beam-wall interface and the point at which the diagonals intersect. New
detailing provisions were introduced in ACI 318-08 to address this issue
(Figure 4-30).
Spacing not to
* exceed 8 in., typical
(b) (c)
Alternate
consecutive
* Spacing
measured
crosstie 90-deg
hooks, both
perpendicular horizontally and
to the axis of
the diagonal
* vertically, typical
bars not to
exceed 14 in., * * Spacing
not to
typical
exceed 8
SECTION SECTION in., typical
Test results are available for coupling beams using the new ACI 318-08
detailing (Wallace, 2007; Naish et al., 2009). Test specimens were one-half
scale, and test geometries and reinforcement detailing were selected to be
representative of common span-to-depth ratios for residential construction
(ln/h=36”/15”=2.4) and office construction (ln/h=60”/18”=3.33). Detailing
for two groups of specimens are shown in Figure 4-31. Maximum expected
shear stresses are approximately vu ,max 6 f c psi and vu ,max 10 f c psi for
span-to-depth ratios of 3.33 and 2.4, respectively.
(a) ACI 318-08 Option 1 (diagonal) (b) ACI 318-08 Option 2 (full)
Figure 4-31 Coupling beam reinforcement detailing (Wallace, 2007).
Test results are presented in Figure 4-32 and Figure 4-33. Rotation levels of
approximately 8% were achieved in all tests prior to any significant strength
degradation. At each aspect ratio, specimens utilizing the two reinforcing
options produced nearly identical force-deformation response (Figure 4-32),
indicating that the transverse reinforcement detailing in Figure 4-31b is as
effective as the detailing in Fig 4-31a.
0 0
Backbone relations derived from tests on beams with ln/h = 2.4, both with
and without slabs, are given in Figure 4-33b. The presence of a slab
increased peak shear strength by 20% to 25%, but did not impact the beam
deformation capacity.
0.3 1.6
ln/h = 2.4
No Slab (B1) 1.4 No Slab
RC Slab (B3) RC Slab
1.2 PT Slab
PT Slab (B4)
0.2 ASCE 41 S.1
V/Vncode
1
Ieff/Ig
0.8
0.6
0.1
0.4
0.2
0 0
0 2 4 6 0 2 4 6 8 10
Rotation [% drift] Rotation [% drift]
(a) Effective stiffness (b) Backbone relations for ln/h=2.4
Figure 4-33 Comparison of: (a) effective stiffness; and (b) backbone relations for coupling
beam test results.
Prior test results reported by Paulay and Binney (1974) were reviewed to
compare effective stiffness measurements. Coupling beams were 6 inches
wide, 31 inches deep, and 40 inches long (ln/h = 1.3). Assuming all
deformations were associated with flexural deformations, effective stiffness
values were Ec I eff 0.08 Ec I g , or about half the values for the ln/h = 2.4 and
ln/h = 3.33 tests discussed above. The difference is likely due to shorter
beam span-to-depth ratios, which result in significant shear deformations.
Yield rotations for the Paulay and Binney (1974) tests are close to the shear
backbone yield relation illustrated in Figure 4-5. If the flexural stiffness is
taken as Ec I eff 0.15 Ec I g based on the longer beam tests, then the
deformations at yield resulting from flexure and shear are equal. This result
is consistent with results reported by Massone (2006) for wall pier tests with
ln/h = 1.1 (i.e., equal flexural and shear deformations at yield). Therefore, for
Ec I eff 0.15 Ec I g , the effective shear stiffness should be taken as
Geff Gc 4 0.4 Ec 4 0.1Ec .
Tests on beams without diagonal reinforcement, with ln/h ratios greater than
3, were conducted by Xiao et al. (1999) and Naish et al. (2009). In this
discussion, “frame beams” refers to beams with standard (horizontal)
longitudinal reinforcement. These tests, as well as others, reveal that “frame
beams” display much more pronounced pinching behavior than diagonally-
reinforced beams. This is especially true where no skin reinforcement is
used. For beams tested with skin reinforcement, total rotations (drift ratios)
at significant shear strength degradation exceeded 4%. The effective flexural
stiffness for “frame beams” with ln/h > 3.0 is also
approximately Ec I eff 0.15 Ec I g .
In summary, yield deformations for coupling beams with ln/h > 2.0 are
dominated by flexure, and use of Ec I eff 0.15 Ec I g and Gc 0.4 Ec are
appropriate. For beams with ln/h < 1.4, deformations due to flexure and
shear are about equal, nonlinear behavior is dominated by shear
deformations, and use of Ec I eff 0.15 Ec I g and Gc 0.1Ec are appropriate.
Linear interpolation of effective stiffness values for clear span-to-depth ratios
1.4 < ln/h < 2.0 is a reasonable approach.
Test results indicate that the use of reduced stiffness values for coupling
beams (e.g., Ec I eff 0.25 Ec I g ) is unlikely to produce excessive cracking or
concrete spalling, either at service level or Maximum Considered Earthquake
(MCE) level analyses. Tests summarized in Naish et al. (2009) indicate
hairline to 1/64” diagonal crack widths, and 1/8” to 3/16” flexural crack
widths at lateral drift levels of 3% to 4% (peak displacement). Residual
crack widths at 4% drift were approximately 1/64” for diagonal cracking, and
1/32” for flexural cracking.
Photos of a test specimen with an aspect ratio of ln/h = 3.33 are provided in
Figure 4-34. Similar crack widths were observed for specimens with smaller
aspect ratios of ln/h = 2.4, even though shear stress levels of 10 fc psi to
14 fc psi were achieved. Substantial pullout of the diagonal bars was
observed for these tests (without strength loss), even for cases where a
reinforced concrete or post-tensioned slab was included.
The moment-hinge model uses rigid plastic rotational springs at each end of
the beam, with the properties shown in Figure 4-36. An effective bending
stiffness of 0.5EcIg was used for both beam aspect ratios.
ln/h = 2.4 ln/h = 3.33
3000 4000
3000
2000
Moment (in-k)
Moment (in-k)
2000
1000
1000
0 0
0 0.07 0.14 0 0.04 0.08 0.12
Rotation (rad) Rotation (rad)
Figure 4-36 Rigid plastic rotational springs for moment-hinge model (half-scale
test specimens).
Use of EcIeff = 0.5EcIg along with the slip/extension spring model was found
to result in an effective flexural stiffness very close to 0.15EcIg. An
50
-50
-100
-100
-200 -150
-0.12 -0.06 0 0.06 0.12 -0.1 -0.05 0 0.05 0.1
Drift (%Rotation) Drift (%Rotation)
(a) Moment-hinge model (ln/h = 2.4) (b) Moment-hinge model (ln/h = 3.33)
Relative Displacement [in]
-4.32 -2.16 0 2.16 4.32
200
ln/h = 2.4
Test
Shear hinge model
100
Lateral Load [k]
-100
-200
-0.12 -0.06 0 0.06 0.12
Rotation [radians]
(c) Shear-hinge model (ln/h = 2.4)
Figure 4-37 Load-deformation relations for moment- and shear-hinge models.
Model and test results presented in Figure 4-37 indicate that both moment-
hinge and shear-hinge models reasonably capture the measured load versus
deformation responses. Accounting for the added flexibility due to slip and
The moment-hinge model does a slightly better job than the shear-
displacement hinge model at representing the shape of the load-displacement
loops at large displacements because the moment-hinge option includes more
variables to control the shape of the hysteretic behavior. It is noted that the
plastic rotation associated with significant loss of lateral load for both beam
aspect ratios was arbitrarily assigned a value of approximately 0.08 based on
the test results. This value substantially exceeds the values for modeling
parameters a = 0.03 and b = 0.05 recommended in ASCE/SEI 41-06.
5’-3”
Core wall
above
hinge zone 51’-5”
Transfer
slabs 5’-3”
Hinge zone
Basement 28”
walls
4’-3”
(a) System configuration (b) Core wall plan section
Figure 4-38 Configuration and plan section of tall core wall building system
used in parametric studies.
The following five cases of relative stiffness were considered. Case 5, with
no factors applied to the uncracked shear stiffness (i.e., Gc = 0.4Ec), was used
to assess the impact of cracking.
Case 1: Stiff diaphragm, with modest stiffness reductions in all elements
Case 2: Soft diaphragm
Case 3: Soft hinge
Case 4: Stiff hinge, with soft diaphragm
Case 5: Uncracked, with no stiffness reductions
40
Case 1
40
Case 3
Case 2 Case 4
No Factors No Factors
Ac*8*(f'c)^1/2 Ac*8*(f'c)^1/2
30 30
Floor Level
Floor Level
20 20
10 10
0 0
20 20
10 10
0 0
-30000 -20000 -10000 0 10000 20000 30000 -30000 -20000 -10000 0 10000 20000 30000
East West Force (K) East West Force (K)
(b) East-west response
Figure 4-39 Variation in shear force over height in the: (a) north-south direction; and (b) east-
west direction, for each case of relative stiffness.
40 Case 1
40
Case 3
Case 2 Case 4
No Factors No Factors
30 30
Floor Level
Floor Level
20 20
10 10
0 0
Figure 4-40 Variation in moment over height in the east-west direction, for
each case of relative stiffness.
To study this result, an alternative model was created with nonlinear fiber
elements provided over the full height of the wall. This model was used to
assess the impact of yielding in the upper levels of the core wall, and to
determine the magnitude and distribution of wall strains at various locations
over the height. A comparison between shear and moment distributions over
height for the base model (fiber hinge) and this alternative model (fiber all) is
provided in Figure 4-41 and Figure 4-42.
30 30
Floor Level
Floor Level
20 20
10 10
0 0
-30000 -20000 -10000 0 10000 20000 30000 -20000 -10000 0 10000 20000
North South Force (K) East West Force (K)
Figure 4-41 Comparison of shear force distribution over height for fiber-hinge and
fiber-all models.
30 30
Floor Level
Floor Level
20 20
10 10
0 0
Figure 4-42 Comparison of moment distribution over height for fiber-hinge and
fiber-all models.
Comparisons between shear and moment distributions over height for the
base model (fiber hinge) and the alternative model (fiber all), considering the
relative stiffness cases identified in Table 4-2, are provided in Figure 4-43
and Figure 4-44. Results again indicate that shear force above the hinge zone
is relatively insensitive to variations in shear stiffness; however, the
40 Case 1 40 Case 1
Case 2 Case 2
No Factors No Factors
Ac*8*(f'c)^1/2 Ac*8*(f'c)^1/2
30 30
Floor Level
Floor Level
20 20
10 10
0 0
Figure 4-43 Comparison of shear force distribution over height for fiber-hinge and fiber-all models,
for each case of relative stiffness.
Case 1 40 Case 1
40
Case 2 Case 2
No Factors No Factors
30 30
Floor Level
Floor Level
20 20
10 10
0 0
Figure 4-44 Comparison of moment distribution over height for fiber-hinge and fiber-all models, for
each case of relative stiffness.
Results for moment distribution over height shown in Figure 4-44 confirm
prior results that yielding in the upper levels substantially reduces moment
The maximum strain values for core wall concrete in compression and core
wall reinforcement in tension over height are plotted below. Distributions of
maximum compression and tension strains for elements along the north wall
of the core are shown in Figure 4-45.
N S
Floor Level
10
N S
Floor Level
10
N S
Floor Level
20
W W W
10
Longitudinal reinforcement over the height of the wall was varied to assess
the effect of shear and flexural strength on response. Maximum longitudinal
reinforcement was provided over the hinge region of the wall in levels 1
through 8. Below the hinge region, wall reinforcement was reduced 30% in
levels B5 to B1. Above the hinge region, wall reinforcement was reduced
18% in levels 9 to 12, 35% in levels 14 to 21, 55% in levels 22 to 31, 67% in
levels 32 to 37, and 82% in levels 38 to 43.
A comparison between shear and moment distributions over height for the
base model (100% Steel) and the alternative model (Reduced Steel) is
provided in Figure 4-47. Changes in the shear and flexural strength of the
100% 100%
40 Steel 40 Steel
Reduced Reduced
Steel Steel
30 30
Floor Level
Floor Level
20 20
10 10
0 0
100% 100%
40 Steel 40 Steel
Reduced Reduced
Steel Steel
30 30
Floor Level
Floor Level
20 20
10 10
0 0
Figure 4-47 Comparison of shear and moment distributions over height for the 100% Steel
and Reduced Steel models.
The effective flexural stiffness of the slab is modeled using slab effective
beam-width models from sources such as Allen and Darvall (1977). In this
model, the centerline panel-to-panel transverse width measured
perpendicular to the direction of loading under consideration, is reduced by
the normalized effective stiffness, , as given in Equation 4-10.
l h3
Ec I effective Ec 2 (4-10)
12
where h is the total slab thickness, and the other parameters are described
below.
for exterior frames loaded parallel to the edge. The effective width given by
Equation 4-10 is applicable for slab-column frame models in which the slab-
beam is modeled as rigid over the width of the column (i.e., the joint region).
Typical values of for interior frames vary from 1/2 to 3/4 for reinforced
concrete construction, and 1/2 to 2/3 for post-tensioned construction. Values
for exterior frames transferring load parallel to the edge are about half of
those for interior connections.
Figure 4-48 shows the normalized effective stiffness, , for interior
connections calculated using Equations 4-10 through 4-12 over a range of
span ratios, l2 / l1. Also shown are typical ranges recommended in the
literature for post-tensioned and reinforced concrete connections. Effective
stiffness values for exterior connections can be estimated as half of the
values shown in the figure.
0.6
l2 / l1=1/3
0.5
PT typical
0.4 0.5 values
0.3
1.0
RC typical
0.2
2.0 values
0.1
3
0
Elastic column
Plastic hinges for slab beams Elastic relation for slab beam
or for torsional element or column
In this model, the column and slab-beam are modeled with concentrated
hinges at each end representing the flexural strengths of the members. The
torsion member is rigid until the connection strength is reached, after which
nonlinear rotation is represented. An advantage of this model is that it
column
M con
M cs
M cs
Figure 4-50 Unbalanced moment transferred between the slab and column
in a torsional connection element.
0.03
0.02
0.01
0.09
Ref: Kang & Wallace, ACI 103(4), 2006 Cyclic load history
Han et al., Mag. of Conc. Research, 58, 2006
0.08 Monotonic load history
Drift Ratio (Total Rotation) at Punching
0.06
0.05
0.04
0.03
0.02
0.01
ACI 318-05 21.11.5 Limit
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Gravity Shear Ratio (Vg /V0), where V0 = (3.5f'c1/2+0.3fpc)bo d
(b) Post-tensioned slab-column connection
Figure 4-51 ASCE/SEI 41-06 Supplement No. 1 modeling parameter a for
reinforced concrete and post-tensioned slab-column
connections (Kang and Wallace, 2006).
Rigid-plastic
hinges
core wall
equivalent
slab-beam
equivalent column
CL
column
Core Wall
CL
B1 B2
l2 l2 l2
CL
l2 B3 B4
l2
CL
0.5l1 0.5l1
Moment Strength
Column
Wall
M M
EIeff 1 EIeff 2
θ θ
A comparison between story drifts for the core wall model and coupled core
wall-slab column model is shown in Figure 4-55. In this case, coupling
between the core walls and the slab-column frame did not significantly
impact story drift in the north-south or east-west directions.
A comparison between column axial stresses for the core wall model and
coupled core wall-slab column model is shown in Figure 4-56. Results are
plotted for two different load cases: (a) 1.2D + 1.6L, and (b) 1.0D + E (for a
single ground motion record).
30 30
Floor Level
Floor Level
20 20
E E
10 10
-1 0 1 -2 0 2
Interstory Drift (%) Interstory Drift (%)
Figure 4-55 Comparison of story drifts in the north-south and east-west directions for the core
wall model and coupled core-slab model.
20 20
E E
10 10
N S N S
0 0
W W
Figure 4-56 Comparison of column axial stress in the north-south and east-west directions for
the core wall model and coupled core-slab model.
Sudden changes in axial stress at levels 9 and 30 are due to changes in the
column cross-section at these levels. Results for this case study building and
the given ground motion suggest that the variation in column axial load due
to coupling between the core wall and the slab-column frame is not
significant relative to the pure gravity load case. This is not necessarily a
general result. For slabs with more longitudinal reinforcement and shorter
spans, a greater variation in column axial load would be expected.
Figure 4-57 Slab-to-wall connection details for Specimen 1 (left) and Specimen 2 (right)
(Klemencic et al., 2006).
Test results are shown in Figure 4-59. Elastic behavior was observed up to a
peak drift ratio of 0.85%, with significant yielding at a drift ratio of
approximately 1.0%.
The biggest difference in performance between the two specimens was the
degree of cracking at the slab-wall interface, shown in Figure 4-60. In
Figure 4-60 Observed cracking at 2.5% drift in Specimen 1 (left) and Specimen 2 (right).
The base of a tall building is often referred to as a podium. Any lower part of
a tall building structure that is larger in floor plate, and contains substantially
increased seismic-force resistance in comparison to the tower above, can be
considered a podium.
Floor and roof slabs are key components in a podium. They act as
diaphragms in shear and flexure, distributing forces to the vertical elements
of the seismic-force-resisting system. Within the diaphragms, collectors,
acting in axial tension and compression, accumulate forces in the diaphragm
and assist in the transfer of forces to walls and frames.
Backstay effects are the transfer of lateral forces from the seismic-force-
resisting elements in the tower into additional elements that exist within the
podium, typically through one or more floor diaphragms. The lateral force
resistance in the podium levels, and force transfer through floor diaphragms
at these levels, helps a tall building resist seismic overturning forces. This
component of overturning resistance is referred to as the backstay effect,
based on its similarity to the back-span of a cantilever beam. It is also
sometimes called “shear reversal” because the shear in the seismic-force-
resisting elements can change direction within the podium levels.
foundation
Force path 1: foundation
V
overturning underneath M
tower core wall
Elevation
Figure A-1 Example of a tall building structural system with a concrete core wall
superstructure and below-grade perimeter retaining walls forming a podium.
Podium and backstay effects are influenced by the type of structural system
in the building. Designing for backstay effects requires careful consideration
Key elements of the podium include the reinforced concrete perimeter walls
at the below-grade levels, floor diaphragms at the below-grade levels, and the
foundations and supporting soils. The most critical diaphragm is the main
backstay diaphragm, which is located at the top of the perimeter walls.
Figure A-2 Construction of a concrete core and below-grade levels of a high-rise building (courtesy of
Magnusson Klemencic Associates).
Not all concrete walls in tall buildings are arranged in a core configuration.
The core arrangement works well for buildings where service functions such
as elevators, stairs, mechanical rooms, and restrooms are located near the
center of the floor plan. Buildings in which these elements are offset, or
buildings with L-shaped or other irregular plan configurations, may need a
series of individual walls or multiple cores, as shown in Figure A-3.
Architectural constraints that affect the location and configuration of
concrete walls apply similarly to the location and configuration of steel
braced frames.
Figure A-3 Construction of concrete walls for a high-rise apartment building. The structural
system has two individual walls, at left, and a concrete core, at right (courtesy of
KPFF).
For the direct load path through the foundation, it is important to consider the
vertical stiffness of the piles or the supporting soil below the foundation. For
the backstay load path, it is important to consider the relative stiffness of the
diaphragms and the perimeter walls, including consideration of horizontal
pressures on the walls, and vertical in-plane rocking resistance below the
walls, provided by the surrounding soil.
The elements in both load paths must have sufficient stiffness and strength to
validate local modeling assumptions. Use of well-designed elements in
redundant load paths is beneficial to the seismic performance of the building,
and can result in an economical design.
The type and configuration of the structural system affects the location and
magnitude of forces that are transferred through the podium. In seismic-
force-resisting systems composed of central cores (e.g., shear walls or braced
frames), the foundation directly below the core is often less stiff than the
backstay load path. Pile foundations, however, tend to be stiffer than mat
slabs, resulting in comparatively less reliance on the backstay load path.
Seismic-force-resisting systems that are more distributed over the building
floor plan tend to have more inherent overturning resistance, and rely less on
the backstay load path.
Generally, the main backstay diaphragm located at the top of the podium
perimeter walls will transfer more force than any other diaphragm. During
preliminary design, it is prudent to assume that this diaphragm will need to
be a structural slab that is significantly thicker than the other floors in the
building, and to plan openings at locations that will not interrupt critical load
paths.
Podium and backstay effects are not limited to tall buildings. Low- and mid-
rise buildings can be subject to the same effects. Similar effects can occur at
any location over the height of a building where lateral elements are
discontinued or reduced in stiffness, such as at building setbacks or step-
backs.
While most tall buildings have configurations that result in podium and
backstay effects, there are exceptions. Examples of building configurations
that will not result in backstay effects include:
Buildings without below-grade levels, or buildings without significantly
increased seismic-force-resistance at the base.
Buildings that extend below grade, but have structural separations
between the superstructure and the podium structure that accommodate
seismic deformations without the transfer of seismic forces.
Buildings with perimeter basement walls, but the walls are located
directly below the seismic-force-resisting elements of the superstructure
above. While there could be a marked change in lateral strength and
stiffness, lateral forces will not be transferred through the floor
diaphragms.
Setbacks or step-backs can also create a strength discontinuity that can result
in concentrated nonlinear behavior in the system at the location of the
setback. In the case of concrete walls, the most desirable nonlinear behavior
is a flexural plastic hinge mechanism at the base of the wall. Setbacks can
diaphragm
tower core wall
collector steel
wall
Section A
tower core wall
collector
wall
A
Elevation
Tall building developments can include two or more towers on the same site,
as shown in Figure A-5. Multiple towers are often founded on a common
base structure, and are likely to differ from each other in mass, stiffness, and
other physical properties.
Nonlinear response history analysis is the best way to evaluate the effects of
multiple towers on a common base, and the transfer forces that can occur in
the connecting floor diaphragms and collectors. In the case of towers with
similar dynamic properties, the potential for unsynchronized movement
should be considered. Although there are no standard criteria for doing so,
one approach would be the application of incoherent ground motions in a
nonlinear response history analysis of the combined structure. Another
approach would include varying the mass, stiffness, and other properties of
the towers, in an attempt to bound the magnitude of potential transfer forces
caused by unsynchronized movement.
In buildings on sloping sites, it is possible for retaining walls on the high side
of the site to extend one or more levels above the walls on the low side of the
site. If connected to the structure at all levels, unsymmetric retaining wall
configurations cause unbalanced lateral resistance and undesirable torsional
response. Sloping sites can also cause unequal horizontal soil pressures
applied to the building, which should be considered in the design of below-
grade floor diaphragms and walls.
For the overall structure to respond as intended, other parts of the seismic
load path (e.g., floor and roof diaphragms, collectors, and connections to the
vertical seismic-force-resisting elements) should experience little or no
inelastic deformation. This philosophy is incorporated into building codes
through the use of the 0 factor, which amplifies design forces to a level that
is intended to approach the capacity of the adjacent vertical seismic-force-
resisting elements in the system. A more direct, and more accurate way to
protect non-yielding elements is to use a capacity design approach as part of
a two-stage design process outlined below.
The first stage of the process determines the strength of yielding elements.
Typically, this is accomplished by designing the building to comply with all
applicable code provisions (except for identified exceptions such as the
height limit). This means that the designated yielding elements are designed
for code-level demands including the R factor. For tall buildings with long
periods, code-level demands are typically governed by minimum base shear
requirements. Alternatively, a code-level analysis could be replaced with an
elastic analysis that is scaled to the service level earthquake.
The second stage of the process is to analyze the structure with a nonlinear
response history analysis using Maximum Considered Earthquake (MCE)
level ground motions. This analysis is intended to:
Nonlinear response history analysis at the MCE level using expected material
properties replaces the application of the code-prescribed overstrength factor,
0, on actions designed to remain elastic. When evaluating actions that are
to remain elastic, the design should consider the dispersion in the nonlinear
response history analysis results, rather than just the average response.
To get appropriate diaphragm and wall forces, the analytical model should
include mass at all levels. In code procedures for calculating equivalent
static lateral story forces, it is often appropriate to neglect mass below grade;
however, this mass should be included in the dynamic analytical model for
backstay effects.
In general, floor diaphragms should be designed for: (1) inertial forces due to
seismic acceleration of the mass tributary to the floor; and (2) seismic forces
that are redistributed between different elements of the seismic-force-
resisting system.
V
M
In the case of very large collector forces, it is often not possible for the
collector element to be concentric with the connecting wall or frame in the
vertical seismic-force-resisting system. Figure A-8 illustrates a collector in
which a portion of the seismic force is transferred directly into the end of the
wall, and the balance is transferred through reinforcing bars placed outside
the wall and shear-friction along the joint between the slab and the wall.
This load path creates an eccentricity between the collector forces and the
reaction within the wall, and creates concentrated levels of diaphragm shear
along the wall. This eccentricity induces additional stresses in the diaphragm
segment adjacent to the wall.
Shear resistance
at wall-to-slab Wall
interface
Collector
reinforcement in
Collector line with vertical
reinforcement element
eccentric to
vertical element
Figure A-8 Eccentric collector and reinforcement into, and alongside, a shear wall.
In reality, all floor and roof diaphragms are semi-rigid because they have a
finite value of in-plane stiffness. However, for practical design purposes, an
idealized rigid or flexible diaphragm is often used to simplify the analysis.
In some cases ASCE/SEI 7-10 requires semi-rigid modeling.
Prior to 2005, building codes did not regulate the choice between diaphragm
modeling assumptions. Beginning with ASCE/SEI 7-05 (ASCE, 2006), use
of a rigid diaphragm assumption for concrete diaphragms required that no
horizontal irregularities be present in the building. Diaphragms in buildings
not meeting this requirement must be modeled as semi-rigid.
where Acv is the net area of concrete section bounded by the slab thickness
and length in the direction of shear force considered. The quantity, 2 f c ,
represents the nominal shear strength of the concrete section, and the
reinforcement ratio, ρn, refers to the reinforcing steel placed parallel to the
direction of shear force considered. For concrete-on-steel deck diaphragms,
the same approach is often used, considering the net thickness of the concrete
topping above the flutes of the deck.
Strut-and-tie models can be used to design diaphragms for shear and flexure.
The design of collectors and diaphragm segments could also be considered in
the strut-and-tie model, if the forces for the whole model are amplified by
0. Code provisions for strut-and-tie models are given in ACI 318-08,
Appendix A.
Typically a floor or roof slab is designed first for gravity loads, then
subsequently checked for in-plane diaphragm forces. The slab reinforcement
utilized for resisting gravity loads should not be considered for resisting in-
plane diaphragm forces. Reinforcement in excess of what is needed to resist
the typical gravity load combinations can be used to resist collector or
diaphragm forces. A portion of all such reinforcement in the top and bottom
of the slab should be continuous, and all lap splices should be Class B.
Code provisions specify the gravity load combinations to be used for design
of the slab reinforcement, including the effect of patterned live loads. When
a slab is designed for gravity loads, it usually has some excess strength due to
Different load factors are specified for gravity loads in combination with
earthquake forces than specified for gravity loads alone. Typically, gravity
loads assumed in combination with earthquake forces are smaller, which is a
second source of excess capacity that can be used to help resist in-plane
diaphragm forces.
When a collector is not placed directly in line with the vertical elements of
the seismic-force-resisting system (see Figure A-8), the eccentricity of the
collector force is countered by in-plane slab moments. Diaphragm segments
around walls (or frames) must be designed for these in-plane moments and
localized increases in diaphragm shear.
Supporting soil/piles – vertical Upper-bound soil Lower-bound soil A fixed base assumption can be used
spring stiffness below perimeter properties properties in lieu of upper-bound properties.
concrete walls
Supporting soil – horizontal Lower-bound soil Upper-bound soil Passive resistance occurs in
spring stiffness on face of properties properties compression but not tension. The
perimeter concrete walls (alternatively soil (will increase overall stiffness of passive resistance can be
springs can be backstay effect, but small compared to the stiffness of the
omitted) will also take force perimeter walls, and thus can often be
out of diaphragms) neglected.
Table A-3 Recommended Stiffness Assumptions for Structural Elements of a Tower and Foundation
Structural element or Assumptions for Assumptions for
property Case 1 Case 2 Notes
Concrete core wall – effective Values recommended in Chapter 4 In typical cases, these stiffness
flexural (EcIeff) and shear (GcA) assumptions are less influential to
stiffness backstay effects and are not bracketed.
Concrete moment frames – Values recommended in Chapter 3 In typical cases, these stiffness
effective flexural (EcIeff)and shear assumptions are less influential to
(GcA) stiffness backstay effects and are not bracketed.
Steel moment frames – Values recommended in Chapter 3 In typical cases, these stiffness
effective flexural (EcIeff) and assumptions are less influential to
shear (GcA) stiffness backstay effects and are not bracketed.
Foundation mat/pile cap – 0.3 times gross section properties, or In typical cases, this stiffness is not
effective flexural stiffness (EcIeff) fully cracked, transformed section influential or uncertain, and need not
properties. be bracketed.
Foundation mat/pile cap – 0.3 times gross section properties, or In typical cases, this stiffness is not
effective shear stiffness (GcA) smaller if shear cracking is expected, influential, and need not be bracketed.
based on shear stress exceeding
3 fc .
Supporting soil/piles – vertical Lower-bound soil Upper-bound soil A fixed base assumption can be used in
spring stiffness properties properties lieu of upper-bound properties.
Passive soil resistance occurs under compression, but not tension. Lateral
soil springs used to represent passive resistance in the model must consider
this behavior. Springs can be compression-only elements, or can be
approximated as tension and compression springs on each side of the below-
grade structure, but modeled with half of the compression stiffness.
The lateral stiffness of passive soil resistance may or may not be important to
model. Often, the stiffness of this load path is small compared to the in-
plane stiffness of the below grade perimeter walls, allowing lateral soil
springs to be omitted.
Upper-bound stiffness properties for lateral soil springs would produce larger
forces in a backstay mechanism overall, but higher passive stiffness will tend
to reduce forces in the below-grade diaphragms and walls, which may not be
conservative. This is particularly true for floor diaphragms in large below
grade structures, where the diaphragm spans are long.
Upper-bound properties for the lateral springs will give larger passive soil
pressures on the perimeter walls, which can govern out-of-plane design of
the walls. Additionally, upper-bound properties for passive soil springs
could govern the shear reversal in a core wall below the main backstay
diaphragm. Appropriate use of upper-bound passive stiffness properties will
depend on the element that is being designed.
Definitions
ACI, 2005, Building Code Requirements for Structural Concrete (ACI 318-
05) and Commentary, (ACI 318R-05), American Concrete Institute,
Farmington Hills, Michigan.
ACI, 2008, Building Code Requirements for Structural Concrete (ACI 318-
08) and Commentary (ACI 318R-08), American Concrete Institute,
Farmington Hills, Michigan.
Adam, C., Ibarra, L.F., and Krawinkler, H., 2004, “Evaluation of P-delta
effects in non-deteriorating MDOF structures from equivalent SDOF
systems,” Proceedings of the 13th World Conference on Earthquake
Engineering, Paper 3407, Vancouver, Canada.
Adebar, P., Ibrahim, A.M.M., and Bryson, M., 2007, “Test of high-rise core
wall: Effective stiffness for seismic analysis,” Structural Journal,
ACI, Vol. 104, No. 5, pp. 549-559.
AISC, 2005a, Prequalified Connections for Special and Intermediate Steel
Moment Frames for Seismic Applications, ANSI/AISC 358-05,
American Institute for Steel Construction, Chicago, Illinois.
AISC, 2005b, Seismic Provisions for Structural Steel Buildings, ANSI/AISC
341-05, American Institute for Steel Construction, Chicago, Illinois.
Aktan, A.E., Bertero, V., and Sakino, K., 1985, “Lateral stiffness
characteristics of reinforced concrete frame-wall structures”,
Structural Journal, ACI, Vol. 86, No. 10, pp. 231-262.
Aktan, A.E., Bertero, V.V., Chowdhury, A.A., Nagashima, T., 1983,
Experimental and Analytical Predictions of the Mechanical
Characteristics of a 1/5-Scale Model of a 7-Story R/C Frame-Wall
Building Structure, UCB/EERC Report 83/13, University of
California, Berkeley, California.
Allen, F.H., and Darvall, P., 1977, “Lateral load equivalent frame,” ACI
Journal Proceedings, Vol. 74, No. 7, pp. 294-299.
Alsiwat, J., and Saatcioglu, M., 1992, “Reinforcement anchorage slip under
monotonic loading,” Journal of Structural Engineering, American
Society of Civil Engineers, Vol. 118, No. 9, pp. 2421-2438.
Helmut Krawinkler
Stanford University
Dept. of Civil and Environmental Engineering
473 Via Ortega
Stanford, California 94305
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PEER 2003/04 Experimental Assessment of Columns with Short Lap Splices Subjected to Cyclic Loads. Murat Melek, John W.
Wallace, and Joel Conte. April 2003.
PEER 2003/03 Probabilistic Response Assessment for Building-Specific Loss Estimation. Eduardo Miranda and Hesameddin
Aslani. September 2003.
PEER 2003/02 Software Framework for Collaborative Development of Nonlinear Dynamic Analysis Program. Jun Peng and
Kincho H. Law. September 2003.
PEER 2003/01 Shake Table Tests and Analytical Studies on the Gravity Load Collapse of Reinforced Concrete Frames. Kenneth
John Elwood and Jack P. Moehle. November 2003.
PEER 2002/24 Performance of Beam to Column Bridge Joints Subjected to a Large Velocity Pulse. Natalie Gibson, André
Filiatrault, and Scott A. Ashford. April 2002.
PEER 2002/23 Effects of Large Velocity Pulses on Reinforced Concrete Bridge Columns. Greg L. Orozco and Scott A. Ashford.
April 2002.
PEER 2002/22 Characterization of Large Velocity Pulses for Laboratory Testing. Kenneth E. Cox and Scott A. Ashford. April
2002.
PEER 2002/21 Fourth U.S.-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced
Concrete Building Structures. December 2002.
PEER 2002/20 Barriers to Adoption and Implementation of PBEE Innovations. Peter J. May. August 2002.
PEER 2002/19 Economic-Engineered Integrated Models for Earthquakes: Socioeconomic Impacts. Peter Gordon, James E.
Moore II, and Harry W. Richardson. July 2002.
PEER 2002/18 Assessment of Reinforced Concrete Building Exterior Joints with Substandard Details. Chris P. Pantelides, Jon
Hansen, Justin Nadauld, and Lawrence D. Reaveley. May 2002.
PEER 2002/17 Structural Characterization and Seismic Response Analysis of a Highway Overcrossing Equipped with
Elastomeric Bearings and Fluid Dampers: A Case Study. Nicos Makris and Jian Zhang. November 2002.
PEER 2002/16 Estimation of Uncertainty in Geotechnical Properties for Performance-Based Earthquake Engineering. Allen L.
Jones, Steven L. Kramer, and Pedro Arduino. December 2002.
PEER 2002/15 Seismic Behavior of Bridge Columns Subjected to Various Loading Patterns. Asadollah Esmaeily-Gh. and Yan
Xiao. December 2002.
PEER 2002/14 Inelastic Seismic Response of Extended Pile Shaft Supported Bridge Structures. T.C. Hutchinson, R.W.
Boulanger, Y.H. Chai, and I.M. Idriss. December 2002.
PEER 2002/13 Probabilistic Models and Fragility Estimates for Bridge Components and Systems. Paolo Gardoni, Armen Der
Kiureghian, and Khalid M. Mosalam. June 2002.
PEER 2002/12 Effects of Fault Dip and Slip Rake on Near-Source Ground Motions: Why Chi-Chi Was a Relatively Mild M7.6
Earthquake. Brad T. Aagaard, John F. Hall, and Thomas H. Heaton. December 2002.
PEER 2002/11 Analytical and Experimental Study of Fiber-Reinforced Strip Isolators. James M. Kelly and Shakhzod M. Takhirov.
September 2002.
PEER 2002/10 Centrifuge Modeling of Settlement and Lateral Spreading with Comparisons to Numerical Analyses. Sivapalan
Gajan and Bruce L. Kutter. January 2003.
PEER 2002/09 Documentation and Analysis of Field Case Histories of Seismic Compression during the 1994 Northridge,
California, Earthquake. Jonathan P. Stewart, Patrick M. Smith, Daniel H. Whang, and Jonathan D. Bray. October
2002.
PEER 2002/07 Seismic Performance of Pile-Wharf Connections. Charles W. Roeder, Robert Graff, Jennifer Soderstrom, and Jun
Han Yoo. December 2001.
PEER 2002/06 The Use of Benefit-Cost Analysis for Evaluation of Performance-Based Earthquake Engineering Decisions.
Richard O. Zerbe and Anthony Falit-Baiamonte. September 2001.
PEER 2002/05 Guidelines, Specifications, and Seismic Performance Characterization of Nonstructural Building Components and
Equipment. André Filiatrault, Constantin Christopoulos, and Christopher Stearns. September 2001.
PEER 2002/04 Consortium of Organizations for Strong-Motion Observation Systems and the Pacific Earthquake Engineering
Research Center Lifelines Program: Invited Workshop on Archiving and Web Dissemination of Geotechnical
Data, 4–5 October 2001. September 2002.
PEER 2002/03 Investigation of Sensitivity of Building Loss Estimates to Major Uncertain Variables for the Van Nuys Testbed.
Keith A. Porter, James L. Beck, and Rustem V. Shaikhutdinov. August 2002.
PEER 2002/02 The Third U.S.-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced
Concrete Building Structures. July 2002.
PEER 2002/01 Nonstructural Loss Estimation: The UC Berkeley Case Study. Mary C. Comerio and John C. Stallmeyer.
December 2001.
PEER 2001/16 Statistics of SDF-System Estimate of Roof Displacement for Pushover Analysis of Buildings. Anil K. Chopra,
Rakesh K. Goel, and Chatpan Chintanapakdee. December 2001.
PEER 2001/15 Damage to Bridges during the 2001 Nisqually Earthquake. R. Tyler Ranf, Marc O. Eberhard, and Michael P.
Berry. November 2001.
PEER 2001/14 Rocking Response of Equipment Anchored to a Base Foundation. Nicos Makris and Cameron J. Black.
September 2001.
PEER 2001/13 Modeling Soil Liquefaction Hazards for Performance-Based Earthquake Engineering. Steven L. Kramer and
Ahmed-W. Elgamal. February 2001.
PEER 2001/12 Development of Geotechnical Capabilities in OpenSees. Boris Jeremi . September 2001.
PEER 2001/11 Analytical and Experimental Study of Fiber-Reinforced Elastomeric Isolators. James M. Kelly and Shakhzod M.
Takhirov. September 2001.
PEER 2001/10 Amplification Factors for Spectral Acceleration in Active Regions. Jonathan P. Stewart, Andrew H. Liu, Yoojoong
Choi, and Mehmet B. Baturay. December 2001.
PEER 2001/09 Ground Motion Evaluation Procedures for Performance-Based Design. Jonathan P. Stewart, Shyh-Jeng Chiou,
Jonathan D. Bray, Robert W. Graves, Paul G. Somerville, and Norman A. Abrahamson. September 2001.
PEER 2001/08 Experimental and Computational Evaluation of Reinforced Concrete Bridge Beam-Column Connections for
Seismic Performance. Clay J. Naito, Jack P. Moehle, and Khalid M. Mosalam. November 2001.
PEER 2001/07 The Rocking Spectrum and the Shortcomings of Design Guidelines. Nicos Makris and Dimitrios Konstantinidis.
August 2001.
PEER 2001/06 Development of an Electrical Substation Equipment Performance Database for Evaluation of Equipment
Fragilities. Thalia Agnanos. April 1999.
PEER 2001/05 Stiffness Analysis of Fiber-Reinforced Elastomeric Isolators. Hsiang-Chuan Tsai and James M. Kelly. May 2001.
PEER 2001/04 Organizational and Societal Considerations for Performance-Based Earthquake Engineering. Peter J. May. April
2001.
PEER 2001/03 A Modal Pushover Analysis Procedure to Estimate Seismic Demands for Buildings: Theory and Preliminary
Evaluation. Anil K. Chopra and Rakesh K. Goel. January 2001.
PEER 2001/02 Seismic Response Analysis of Highway Overcrossings Including Soil-Structure Interaction. Jian Zhang and Nicos
Makris. March 2001.
PEER 2001/01 Experimental Study of Large Seismic Steel Beam-to-Column Connections. Egor P. Popov and Shakhzod M.
Takhirov. November 2000.
PEER 2000/10 The Second U.S.-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced
Concrete Building Structures. March 2000.
PEER 2000/08 Behavior of Reinforced Concrete Bridge Columns Having Varying Aspect Ratios and Varying Lengths of
Confinement. Anthony J. Calderone, Dawn E. Lehman, and Jack P. Moehle. January 2001.
PEER 2000/07 Cover-Plate and Flange-Plate Reinforced Steel Moment-Resisting Connections. Taejin Kim, Andrew S. Whittaker,
Amir S. Gilani, Vitelmo V. Bertero, and Shakhzod M. Takhirov. September 2000.
PEER 2000/06 Seismic Evaluation and Analysis of 230-kV Disconnect Switches. Amir S. J. Gilani, Andrew S. Whittaker, Gregory
L. Fenves, Chun-Hao Chen, Henry Ho, and Eric Fujisaki. July 2000.
PEER 2000/05 Performance-Based Evaluation of Exterior Reinforced Concrete Building Joints for Seismic Excitation. Chandra
Clyde, Chris P. Pantelides, and Lawrence D. Reaveley. July 2000.
PEER 2000/04 An Evaluation of Seismic Energy Demand: An Attenuation Approach. Chung-Che Chou and Chia-Ming Uang. July
1999.
PEER 2000/03 Framing Earthquake Retrofitting Decisions: The Case of Hillside Homes in Los Angeles. Detlof von Winterfeldt,
Nels Roselund, and Alicia Kitsuse. March 2000.
PEER 2000/02 U.S.-Japan Workshop on the Effects of Near-Field Earthquake Shaking. Andrew Whittaker, ed. July 2000.
PEER 2000/01 Further Studies on Seismic Interaction in Interconnected Electrical Substation Equipment. Armen Der Kiureghian,
Kee-Jeung Hong, and Jerome L. Sackman. November 1999.
PEER 1999/14 Seismic Evaluation and Retrofit of 230-kV Porcelain Transformer Bushings. Amir S. Gilani, Andrew S. Whittaker,
Gregory L. Fenves, and Eric Fujisaki. December 1999.
PEER 1999/13 Building Vulnerability Studies: Modeling and Evaluation of Tilt-up and Steel Reinforced Concrete Buildings. John
W. Wallace, Jonathan P. Stewart, and Andrew S. Whittaker, editors. December 1999.
PEER 1999/12 Rehabilitation of Nonductile RC Frame Building Using Encasement Plates and Energy-Dissipating Devices.
Mehrdad Sasani, Vitelmo V. Bertero, James C. Anderson. December 1999.
PEER 1999/11 Performance Evaluation Database for Concrete Bridge Components and Systems under Simulated Seismic
Loads. Yael D. Hose and Frieder Seible. November 1999.
PEER 1999/10 U.S.-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete
Building Structures. December 1999.
PEER 1999/09 Performance Improvement of Long Period Building Structures Subjected to Severe Pulse-Type Ground Motions.
James C. Anderson, Vitelmo V. Bertero, and Raul Bertero. October 1999.
PEER 1999/08 Envelopes for Seismic Response Vectors. Charles Menun and Armen Der Kiureghian. July 1999.
PEER 1999/07 Documentation of Strengths and Weaknesses of Current Computer Analysis Methods for Seismic Performance of
Reinforced Concrete Members. William F. Cofer. November 1999.
PEER 1999/06 Rocking Response and Overturning of Anchored Equipment under Seismic Excitations. Nicos Makris and Jian
Zhang. November 1999.
PEER 1999/05 Seismic Evaluation of 550 kV Porcelain Transformer Bushings. Amir S. Gilani, Andrew S. Whittaker, Gregory L.
Fenves, and Eric Fujisaki. October 1999.
PEER 1999/04 Adoption and Enforcement of Earthquake Risk-Reduction Measures. Peter J. May, Raymond J. Burby, T. Jens
Feeley, and Robert Wood.
PEER 1999/03 Task 3 Characterization of Site Response General Site Categories. Adrian Rodriguez-Marek, Jonathan D. Bray,
and Norman Abrahamson. February 1999.
PEER 1999/02 Capacity-Demand-Diagram Methods for Estimating Seismic Deformation of Inelastic Structures: SDF Systems.
Anil K. Chopra and Rakesh Goel. April 1999.
PEER 1999/01 Interaction in Interconnected Electrical Substation Equipment Subjected to Earthquake Ground Motions. Armen
Der Kiureghian, Jerome L. Sackman, and Kee-Jeung Hong. February 1999.
PEER 1998/08 Behavior and Failure Analysis of a Multiple-Frame Highway Bridge in the 1994 Northridge Earthquake. Gregory L.
Fenves and Michael Ellery. December 1998.
PEER 1998/07 Empirical Evaluation of Inertial Soil-Structure Interaction Effects. Jonathan P. Stewart, Raymond B. Seed, and
Gregory L. Fenves. November 1998.
PEER 1998/06 Effect of Damping Mechanisms on the Response of Seismic Isolated Structures. Nicos Makris and Shih-Po
Chang. November 1998.
PEER 1998/04 Pacific Earthquake Engineering Research Invitational Workshop Proceedings, May 14–15, 1998: Defining the
Links between Planning, Policy Analysis, Economics and Earthquake Engineering. Mary Comerio and Peter
Gordon. September 1998.
PEER 1998/03 Repair/Upgrade Procedures for Welded Beam to Column Connections. James C. Anderson and Xiaojing Duan.
May 1998.
PEER 1998/02 Seismic Evaluation of 196 kV Porcelain Transformer Bushings. Amir S. Gilani, Juan W. Chavez, Gregory L.
Fenves, and Andrew S. Whittaker. May 1998.
PEER 1998/01 Seismic Performance of Well-Confined Concrete Bridge Columns. Dawn E. Lehman and Jack P. Moehle.
December 2000.
PEER 2010/111 Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings. October 2010.
PEER 2010/109 Report of the Seventh Joint Planning Meeting of NEES/E-Defense Collaboration on Earthquake Engineering.
Held at the E-Defense, Miki, and Shin-Kobe, Japan, September 18–19, 2009. August 2010.
PEER 2010/107 Performance and Reliability of Exposed Column Base Plate Connections for Steel Moment-Resisting Frames.
Ady Aviram, Božidar Stojadinovic, and Armen Der Kiureghian. August 2010.
PEER 2010/106 Verification of Probabilistic Seismic Hazard Analysis Computer Programs. Patricia Thomas, Ivan Wong, and
Norman Abrahamson. May 2010.
PEER 2010/105 Structural Engineering Reconnaissance of the April 6, 2009, Abruzzo, Italy, Earthquake, and Lessons Learned. M.
Selim Günay and Khalid M. Mosalam. April 2010.
PEER 2010/104 Simulating the Inelastic Seismic Behavior of Steel Braced Frames, Including the Effects of Low-Cycle Fatigue.
Yuli Huang and Stephen A. Mahin. April 2010.
PEER 2010/103 Post-Earthquake Traffic Capacity of Modern Bridges in California. Vesna Terzic and Božidar Stojadinović. March
2010.
PEER 2010/102 Analysis of Cumulative Absolute Velocity (CAV) and JMA Instrumental Seismic Intensity (IJMA) Using the PEER–
NGA Strong Motion Database. Kenneth W. Campbell and Yousef Bozorgnia. February 2010.
PEER 2010/101 Rocking Response of Bridges on Shallow Foundations. Jose A. Ugalde, Bruce L. Kutter, Boris Jeremic
PEER 2009/109 Simulation and Performance-Based Earthquake Engineering Assessment of Self-Centering Post-Tensioned
Concrete Bridge Systems. Won K. Lee and Sarah L. Billington. December 2009.
PEER 2009/108 PEER Lifelines Geotechnical Virtual Data Center. J. Carl Stepp, Daniel J. Ponti, Loren L. Turner, Jennifer N.
Swift, Sean Devlin, Yang Zhu, Jean Benoit, and John Bobbitt. September 2009.
PEER 2009/107 Experimental and Computational Evaluation of Current and Innovative In-Span Hinge Details in Reinforced
Concrete Box-Girder Bridges: Part 2: Post-Test Analysis and Design Recommendations. Matias A. Hube and
Khalid M. Mosalam. December 2009.
PEER 2009/106 Shear Strength Models of Exterior Beam-Column Joints without Transverse Reinforcement. Sangjoon Park and
Khalid M. Mosalam. November 2009.
PEER 2009/105 Reduced Uncertainty of Ground Motion Prediction Equations through Bayesian Variance Analysis. Robb Eric S.
Moss. November 2009.
PEER 2009/104 Advanced Implementation of Hybrid Simulation. Andreas H. Schellenberg, Stephen A. Mahin, Gregory L. Fenves.
November 2009.
PEER 2009/103 Performance Evaluation of Innovative Steel Braced Frames. T. Y. Yang, Jack P. Moehle, and Božidar
Stojadinovic. August 2009.
PEER 2009/102 Reinvestigation of Liquefaction and Nonliquefaction Case Histories from the 1976 Tangshan Earthquake. Robb
Eric Moss, Robert E. Kayen, Liyuan Tong, Songyu Liu, Guojun Cai, and Jiaer Wu. August 2009.
PEER 2009/101 Report of the First Joint Planning Meeting for the Second Phase of NEES/E-Defense Collaborative Research on
Earthquake Engineering. Stephen A. Mahin et al. July 2009.
PEER 2008/104 Experimental and Analytical Study of the Seismic Performance of Retaining Structures. Linda Al Atik and Nicholas
Sitar. January 2009.
PEER 2008/103 Experimental and Computational Evaluation of Current and Innovative In-Span Hinge Details in Reinforced
Concrete Box-Girder Bridges. Part 1: Experimental Findings and Pre-Test Analysis. Matias A. Hube and Khalid M.
Mosalam. January 2009.
PEER 2008/102 Modeling of Unreinforced Masonry Infill Walls Considering In-Plane and Out-of-Plane Interaction. Stephen
Kadysiewski and Khalid M. Mosalam. January 2009.
PEER 2008/101 Seismic Performance Objectives for Tall Buildings. William T. Holmes, Charles Kircher, William Petak, and Nabih
Youssef. August 2008.
PEER 2007/101 Generalized Hybrid Simulation Framework for Structural Systems Subjected to Seismic Loading. Tarek Elkhoraibi
and Khalid M. Mosalam. July 2007.