Displacement-Based Design of Cantilever Bridge Piers Matthew J. Tobolski, a) M.EERI, and José I. Restrepo, b) M.EERI Performance-based seismic design requires the development of methods that address a number of explicitly defined performance objectives. Displacement- based design methods are generally recognized as excellent candidates for use within a performance-based design framework due to the ability to predict structural damage states. This paper presents a two-level displacement-based design method for use in the design of bridge piers. The method considers two performance objectives: immediate operation and life-safety. This method is formulated in a probabilistic framework to allow for the explicit consideration of uncertainty associated with seismic design. Results of the method presented indicate that for many common situations the life-safety performance objective will control the design. However, it is possible for immediate operation considerations to control if strict residual drift limits are imposed. Also, it was shown that for tall columns the flexural design moment is fairly independent of the column diameter. This procedure is intended to provide bridge engineers with a simple and transparent seismic design tool. INTRODUCTION Performance-based seismic design requires the development of methods that address a number of explicitly defined performance objectives (SEAOC, 1999). Displacement-based design methods are generally recognized as excellent candidates for use within a performance-based design framework due to the ability to predict structural damage states. A variety of displacement-based design methods have been developed in recent years in an attempt to meet the overarching goals of performance-based seismic design (Sullivan et al., 2003). Significant variations in methods and design considerations have been observed a) Graduate Student Researcher, University of California San Diego, Department of Structural Engineering, 9500 Gilman Dr. MC 0085, La Jolla, CA 92093-0085, [email protected]b) Associate Professor, University of California San Diego, Department of Structural Engineering, 9500 Gilman Dr. MC 0085, La Jolla, CA 92093-0085, [email protected]
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Displacement-Based Design of Cantilever Bridge Piers
Matthew J. Tobolski, a) M.EERI, and José I. Restrepo,b) M.EERI
Performance-based seismic design requires the development of methods that
address a number of explicitly defined performance objectives. Displacement-
based design methods are generally recognized as excellent candidates for use
within a performance-based design framework due to the ability to predict
structural damage states. This paper presents a two-level displacement-based
design method for use in the design of bridge piers. The method considers two
performance objectives: immediate operation and life-safety. This method is
formulated in a probabilistic framework to allow for the explicit consideration of
uncertainty associated with seismic design. Results of the method presented
indicate that for many common situations the life-safety performance objective
will control the design. However, it is possible for immediate operation
considerations to control if strict residual drift limits are imposed. Also, it was
shown that for tall columns the flexural design moment is fairly independent of
the column diameter. This procedure is intended to provide bridge engineers with
a simple and transparent seismic design tool.
INTRODUCTION
Performance-based seismic design requires the development of methods that address a
number of explicitly defined performance objectives (SEAOC, 1999). Displacement-based
design methods are generally recognized as excellent candidates for use within a
performance-based design framework due to the ability to predict structural damage states. A
variety of displacement-based design methods have been developed in recent years in an
attempt to meet the overarching goals of performance-based seismic design (Sullivan et al.,
2003). Significant variations in methods and design considerations have been observed
a) Graduate Student Researcher, University of California San Diego, Department of Structural Engineering, 9500 Gilman Dr. MC 0085, La Jolla, CA 92093-0085, [email protected] b) Associate Professor, University of California San Diego, Department of Structural Engineering, 9500 Gilman Dr. MC 0085, La Jolla, CA 92093-0085, [email protected]
between these procedures, with the main similarity being the goal of restricting structural
displacements.
Displacement-based design methods began emerging in the early 1990’s as a means to
design structures through a more rational means (Moehle, 1992; Kowalsky et. al, 1994; Calvi
and Kingsley, 1995). The method proposed by Moehle (1992) relies on estimates of
structural period and strength to determine displacement and curvature demands. These
values are then compared with system capacities to ensure a given performance objective is
attained. Kowalsky, et. al (1994) proposed a method in which a target drift is selected and the
required stiffness is determined based on a substitute structure which is related to the
structure’s ultimate displaced state. Calvi and Kingsley (1995) developed a similar
displacement-based design approach that was extended for use on multiple degree-of-
freedom systems based on an assumed displaced shape of the structure. Aschheim and Black
(2000) presented a method which utilizes yield point spectra relating yield force and
displacement for various ductility levels in order to achieve desired deformation states. This
method then used conventional force based procedures to allocate lateral resistance in a
structure. Browning (2001) proposed a method in which a target drift is specified and design
is carried out with the goal of achieving specified deformation limits. However, in this
method there is no specified limit to member rotation and ductility and consequently no
direct control over damage limit states. The design procedure provided by SEAOC (1999)
similarly specifies a target drift without consideration for the ductility demands imposed on
the structure. Chopra and Goel (2001) have presented a design method which utilizes an
inelastic design spectrum combined with specified drift and rotation limits, thus explicitly
considering inelastic action and damage limit states.
While some procedures consider multiple performance objectives, many of these
procedures consider the life safety objective as the sole criterion. Emerging performance-
based design methods dictate that immediate operation must also be considered during a
seismic design procedure as a means to meet societal performance expectations (FEMA,
1997). Therefore, for a displacement based design method to be used in conjunction with
performance-based design, it must also properly consider multiple performance objectives. A
two-level displacement design method is presented in this paper for use with bridge piers that
can be treated as single-degree of freedom; however this procedure can easily be extended to
include a variety of other performance objectives. The method discussed herein is presented
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as a framework for use in reliability based design as uncertainties in seismic design are
explicitly considered. This method assumes initial member sizing has been performed such
that an initial column diameter has been previously determined. For this paper, foundation
flexibility is neglected, but this procedure can be easily extended to consider the effects of
flexible boundary conditions.
The two performance objectives that are considered in this paper are immediate operation
and life safety. The goal of the immediate operation performance objective is to control
structural damage and residual displacements such that an aerial structure can be reopened
shortly after a moderately strong seismic event with little to no repair, and minimal
interruption to traffic flow. In the method presented herein, longitudinal reinforcement tensile
and concrete compressive strains and residual displacements are limited in order to meet this
performance requirement. Earthquake demands from a 50% probability of exceedance in 50
years are considered appropriate for this performance objective in this paper (50/50).
The life safety performance objective implies a structure will be brought significantly
beyond the elastic limit during a rare but strong intensity ground motion. Significant inelastic
response and damage, as well as development of life-hazardous conditions, but no collapse,
are expected in parts of the structure. Typically, regions of inelastic response develop in the
form of flexural plastic hinges. These regions must be recognized in the design stage and be
detailed through capacity design to ensure the chosen inelastic mechanism is able to form and
be maintained (Priestley et al., 1996). In the case of the bridge pier discussed in this paper,
this mechanism comes as the development of a flexural plastic hinge at the column base.
Limits on concrete compressive and longitudinal reinforcement tensile strains are used to
determine reliable system displacement limits. Demands for this performance objective are
based on a 2% probability of exceedance in 50 year seismic event (2/50).
DEVELOPMENT OF SYSTEM CAPACITY AND DEMANDS
PROBABILISTIC FRAMEWORK
In seismic design, much the same as in all structural design, there is uncertainty in both
structural demands and capacities. A fraction of the uncertainty is epistemic in nature and
could be reduced with future knowledge. In terms of demand, the exact nature of future
seismic actions a structure may undergo is not known during design and consequently in
practice structures must be designed to some derived demand. In terms of capacity, the
displacement capacity corresponding to a given damage limit state of a reinforced concrete
member cannot be determined exactly. In seismic design this is even truer as a member’s
displacement capacity is also a function of the demand due to the progressive formation of
damage during excitation. Furthermore, the uncertainty associated with structural behavior
comes not only in the form of variability in material properties and construction quality, but
also time dependant variations due to such phenomenon as strain rate and strain aging
(Restrepo et al., 1994) and in some special cases due to low temperature (Suleiman et al.,
2006), creep and shrinkage. The uncertainty in both potential seismic actions and structural
behavior exemplifies the need for seismic design of structures to be performed in the context
of reliability based procedures. The method present in this paper is developed within a
framework such that the seismic design can be performed in a probabilistic context. There are
a variety of factors presented in this paper that are intended to address uncertainties in design;
however, significant effort is required to calibrate these values.
DESIGN SPECTRUM
An underlying assumption of the proposed design method is that reasonable elastic design
spectra are employed. The design method presented herein assumes a design spectrum which
is characterized by a constant acceleration, constant velocity, and constant displacement
regions. A representative displacement design spectrum consistent with these parameters is
shown in Figure 1. Design spectra produced through NEHRP, Eurocode 8, and proposed
NCHRP/AASHTO provisions have these characteristics with the exception that Eurocode 8
also includes a fourth region for long-period structures in which the spectral displacement is
equivalent to the peak ground displacement (FEMA, 1997; CEN, 1998; Imbsen, 2006).
NEHRP and NCHRP/AASHTO provisions indicate that the constant displacement region
begins at a significantly longer period as compared to the Eurocode 8 provisions. A
comprehensive study on the characteristics of long period displacement response spectra has
shown that there are a variety of factors influencing the location and relative magnitude of
the constant displacement regions (Faccioli et al., 2003). The significant difference between
the various design provisions combined with the results of the aforementioned study indicate
a lack consistency in codified earthquake characteristics and a general lack of understanding
of the nature of seismic demands.
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In a general sense, design spectra are developed through review of earthquake ground
records, an assessment of potential source mechanisms, attenuation laws, structural damping
assumptions and local site conditions. The seismic demands used in practice represent some
probabilistic demand level based on current state-of-knowledge. As the displacement based
method presented herein is intended to be used in a probabilistic context, the uncertainty in
the seismic demands must be unambiguously considered. Consequently, probability of
exceedance for a given spectrum may not be appropriate for achieving an overall reliability
desired. In practice, it is common to consider the mean value of seismic demand for design
(Abrahamson and Bommer, 2005). It is possible that a different probabilistic demand level
may be desirable and for the purposes of the proposed method, a scaling factor, analogous to
the load factor in the Load and Factor Resistance Design (LFRD) in the design for gravity
loading, is introduced to account for uncertainties in the design seismic demand and to
modify a given spectrum to some other probabilistic level. This factor, CQ, will be applied to
the elastic design spectrum to modify the spectral values as desired. It highly likely that this
modification factor be period dependant as the current understanding of earthquake ground
motions creates a situation where the level of uncertainty is not constant for all structural
periods (Crowley et. al, 2005). In the event that site-specific spectra are developed, the design
spectrum can be generated such that the values represent a given level of reliability resulting
in an uncertainty factor equal to unity.
DAMPING
Many displacement-based design methods use the concept of equivalent viscous damping
in developing the displacement demands on a system (Chopra and Goel, 2001; Priestley and
Kowalsky, 2000; SEAOC, 1999). A major assumption used in these cases is that hysteretic
damping can be converted to viscous damping based on characteristics of a quasi-static
hysteretic relationship. Concerns regarding this assumption relate to the nature of viscous
damping, which is velocity dependant, unlike hysteretic damping which relies on large
inelastic displacement cycles. Furthermore, the quantification of hysteretic damping based on
static testing implies the system undergoes full displacement cycles constantly, which is
unlikely to occur during actual earthquake excitation. Increasing the level of damping based
on equivalent viscous damping may lead to an underestimation of seismic displacement
demands and misrepresentation of system response unless statistically based correction
factors are implemented in the design procedure.
For the purposes of this procedure, the authors propose the use of two damping ratios
based on the seismic design level without consideration of equivalent viscous damping.
Damping ratios of 2% and 5% are recommended for the 50/50 and 2/50 level events,
respectively. The lower damping ratio for the immediate operation level is intended to reflect
the lower amount of inelastic action resulting from the seismic demands associated with this
performance objective. As most design spectra are provided for a damping ratio equal to 5%,
a modification factor is needed to alter the spectral values based on the desired damping
level. A variety of methods have been developed to modify the design spectra based on
damping (Kawashima and Aizawa, 1986; FEMA, 1997; Fu and Cherry, 1999, NCHRP,
2001). An appropriate model must be selected in calibrating this design method.
INELASTIC RESPONSE
Both immediate operation and life safety performance objectives are expected to produce
inelastic structural response during seismic actions. Consequently, the elastic spectral
displacements which are used must be modified to account for inelasticity. Many design
procedures currently in use in the United States are based on the “Equal Displacement
Concept”, that is, the inelastic displacement demand is the equal to the elastic demand
(Caltrans, 2004; AASHTO 2004). This relationship stems back to a study conducted by
Veletsos and Newmark (1960) in which the El Centro ground motion record was analyzed for
a variety of inelastic and elastic systems. Results from that study indicated that for low to
medium frequencies the elastic and inelastic displacements are nearly equal and for higher
frequencies the “Equal Energy Concept” was valid. Many studies have shown that the “Equal
Displacement Concept” provides an acceptable estimate for median inelastic displacement
demands for longer periods but is not valid in short period structures (Chopra, 2001; Miranda
and Bertero, 1994). Most importantly, it has been shown that this concept only represents
median response with no consideration for dispersion, which can be significant (Ruiz-Garcia
and Miranda, 2003). Understanding the short-period displacement amplification, Caltrans
(1999) recommends that designers modify the displacement demand for structures with
periods less than 0.7 seconds but provides no insight into the magnitude of this modification
factor. Proposed NCHRP/AASHTO design provisions also include provisions to account for
the increase in displacement demands due to inelastic actions (Imbsen, 2006). The design
method presented in this paper takes into consideration the inelastic displacement
magnification and associated dispersion over all periods in a single formulation.
For the proposed design method, the elastic displacement demand taken from design
spectra are modified by the inelastic displacement ratio, RC , to provide an estimate of the
inelastic displacement demand. The inelastic displacement ratio is defined as:
iR
e
C Δ=Δ
(1)
where iΔ is the inelastic displacement demand and eΔ is the elastic spectral displacement
demand.
Values of the inelastic displacement ratio cannot be determined in closed-form for
general earthquake excitation and must be calculated through a series of non-linear time-
history analyses. Ruiz-Garcia and Miranda (2003) performed a series of analyses on single
degree-of-freedom systems with constant damping to quantify this value over a variety of
parameters when subjected to far field ground motions. A similar study was performed in
developing the design method presented herein except tangent stiffness damping was
considered. Both studies indicate that there is a significant amount of variation in the inelastic
displacement ratio due to the characteristics of the input ground motion record. Figure 2
shows the basic trends associated with the inelastic displacement ratio as compared to the
fundamental period. Results from the authors’ investigation were used to develop the
following relationship for the inelastic displacement ratio:
0.5
0.3
1 11.7RC
TμΔ −
= +⋅
(2)
where μΔ is the displacement ductility and T is the structural period. This relationship
represents the 90th percentile of results. This percentile is determined based on the apparent
log-normal distribution of inelastic displacement demand for a given period. The relationship
is valid for displacement ductility values up to 8 with structural periods larger than 0.3
seconds located on firm soils.
As multiple damping ratios are proposed for this design method, the influence of damping
on inelastic response was also investigated. Results indicate that a reduction in the tangent
stiffness viscous damping ratio will lead to a reduction in the inelastic displacement ratio for
all periods. For the purposes of the proposed design method, the same inelastic displacement
relationship will be used for both performance objectives, resulting in a slightly higher
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mass-proportional?
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do you want to say that the analysis were conducted with 2 and 5% stiffness-proportional damping???
percentile of inelastic displacement demands for the immediate operation performance
objective.
The relationship presented in Equation 2 was developed based on a series of non-linear
analyses of single degree-of-freedom systems subjected to far field ground motion records on
firm soils. To determine the effectiveness of the proposed relationship for near fault events, a
series of analyses were conducted on non-linear single degree-of-freedom oscillators
subjected to ground motion records with near field characteristics. The ground motion
records used for this investigation were selected based on directivity and fling-step
characteristics observed in the records. Results from these analyses indicate that directivity
effects can lead to more severe inelastic displacement demands and the proposed relationship
may severely underestimate inelastic seismic demands. Consequently, the proposed equation
is only recommended for structures located where near fault ground motion characteristics
are not anticipated. Future efforts are needed to develop appropriate inelastic displacement
relationships for near field actions.
COMBINATION OF DISPLACEMENT AMPLIFICATION FACTORS
Both the ground motion uncertainty and the inelastic displacement factors must be
applied to the elastic spectral displacement demand to provide an appropriate design
displacement demand. The likelihood that a given input ground motion will result in both an
elastic displacement demand that is larger than the design value and an inelastic displacement
ratio greater than that provided by Equation 2 is small. Consequently, an absolute sum of the
modifications factors may be overly conservative. A more reasonable combination is using a
square-root-sum-of-squares relationship to create a generalized modification factor:
2 2( 1) ( 1) 1Q RC C CΔ = − + − + (3)
A key objective of this paper is to present a framework for reliability-based displacement-
based design. Consequently, future studies may determine other factors need to be applied to
the design ordinate obtained from an elastic design spectrum. Therefore, it is more
appropriate to present this generalized modification factor as a combination of all factors
deemed appropriate for considerations as follows:
2( 1) 1ii
C CΔ = − +∑ (4)
where 1iC ≥ and represents any factor used to modify the elastic design displacement.
The resulting design inelastic seismic displacement demand in the constant acceleration
and constant velocity regions is defined as:
u C TαΔΔ = ⋅ ⋅ (5)
with α defined as the slope of the elastic displacement response spectrum in the constant
velocity region for a given performance objective (see Figure 1).
DISPLACEMENT CAPACITY
For the design method presented herein, displacement limits corresponding to a given
damage limit state are calculated based on curvature relationships. Displacement limits for a
member deformed beyond the elastic limit are determined by separating the yield and plastic
displacements. The yield displacement is determined considering a linear variation of
curvature from the center of mass of the superstructure to the idealized yield curvature at the
base of the column. The assumed curvature profile at yield is shown in Figure 3a. The
idealized yield curvature can be determined using the relationship developed by Priestley
(2003):
yy D
λ εφ
⋅= (6)
where λ is a section shape factor, yε is the yield strain of reinforcing steel, and D is the
diameter of the column. The related yield displacement for a column of idealized height, h,
is:
2
3y
y
hφ ⋅Δ = (7)
The yield displacement defined from Equation 7 assumes a column on an infinitely rigid
foundation, an assumption which is not acceptable for most bridge applications. Considering
foundation flexibility results in larger yield displacements and can play a significant role in
this displacement design method. If assuming the column behaves elastic-perfectly plastic,
the displacement resulting from flexibility in the foundation is a function of the flexural
design moment, Mdesign, which is an end result of this procedure. This necessitates an initial
assumption and subsequent iteration of the procedure in order to converge on the final
flexural design moment. Foundation flexibility is not explored in the current paper;
consequently the foundation is idealized with perfect fixity.
Beyond the elastic limit, the displacement capacity is derived from a combination of yield
and plastic curvatures. The assumed curvature profile at the ultimate state is presented in
Figure 3b. The plastic curvature can be determined based on a specified ultimate curvature
ductility with the plastic curvature:
( )1p yφφ μ φ= − ⋅ (8)
Using the defined curvature profile, the plastic displacement is calculated as:
2p
p p p
ll hφ
⎛ ⎞Δ = ⋅ ⋅ −⎜ ⎟
⎝ ⎠ (9)
where lp is the idealized plastic hinge length. For the life safety performance objective, where
curvature ductility values greater than 5 are anticipated, the plastic hinge length can be take
as one-half the column diameter or it can be calculated based on other recommendations such
as those by Priestley et al. (1996) or Hines et al. (2004). While these references provide fairly
deterministic values for the equivalent plastic hinge length, there can be significant variation
in the actual spread of plasticity. This dispersion can relate to material properties, axial load
effects, and column detailing.
While specifying a value for the curvature ductility provides a means to calculate the
ultimate displacement capacity and affords some level of insight into damage in a member, it
does not directly relate damage and displacement. Strain level in a member offers a more
rational means to predict damage in a reinforced concrete member. Consequently, it is
beneficial to relate the strain state to curvature ductility. Properties of a given section and
axial loading affect the strain-ductility relationship. In order to develop relationships between
strain and curvature ductility, a series of moment-curvature analyses are required. An
example result of this type of analysis is presented in Figure 4, where two moment curvature
analyses were performed for a given column configuration subjected to two different axial
loads.
From the moment-curvature results, strain states are identified and relationships between
axial force and curvature ductility can be developed. For this example three damage states are
considered which represent certain strain states in the concrete and steel. Damage State I
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,and hence, on the equivalent plastic hinge length (add also the paper by Stephan, Restrepo, Seible and Schoettler...)
represents minimal damage in a reinforced concrete member, and is associated with a strain
in unconfined concrete of -0.004 or strain in longitudinal reinforcing steel of 0.01.
Damage State II represents a state in which a section is slightly damaged but repairable,
and is based on incipient spalling of the concrete cover when confined concrete strain equals
-0.004 or when longitudinal reinforcing steel begins to buckle due to cyclic loading. The
condition which leads to cyclic bar buckling can be adopted from the work of Rodriguez et
al. (1999) or from the following simple relationship:
4143
100 2b su
s c
sd εε ε
−− = ≤ (10)
where sε and cε are the strains in steel and concrete, respectively, at the location of extreme
longitudinal reinforcement under a single displacement cycle, s is the transverse
reinforcement spacing, bd is the diameter of the longitudinal reinforcement and suε is the
tensile strain at the peak axial stress of the longitudinal reinforcement. The value of cε must
be less than or equal to the spalling strain of the concrete cover (i.e. the cover must have
spalled).
Damage State III represents the ultimate curvature state defined by the crushing of the
confined concrete core or the cyclic fracture of reinforcing steel. An in depth series of
analyses are required to develop equations relating given strain states to curvature ductility.
Table 1 provides a comparison of these three damage states with the seismic demand and
performance levels as considered in this paper.
While the previous procedure provides a reasonable method for determining displacement
capacity of a member based on curvature, there is inherent variability in the actual
displacement capacity due to material variation, construction quality, and relationships
between strain and a given damage state. To address the uncertainty in actual displacement
capacity, a reduction factor is applied to the plastic displacement resulting in what is
classified as the reliable ultimate displacement capacity. The yield displacement will be
assumed valid without modification. The resulting ultimate displacement capacity is:
u y DS pηΔ = Δ + Δ (11)
where DSη is the plastic displacement reduction factor for a given damage state. This factor is
analogous to the strength reduction factor employed in LFRD. The value for the reduction
factor is expected to be different for each damage state due to the lack of understanding of
the behavior of reinforced concrete members to earthquake demands at various damage
states.
DESIGN PROCEDURE
The design method presented herein is intended for use with single degree-of-freedom
systems as shown in Figure 5 that can be characterized by elastic-perfectly plastic response.
Only flexural deformation modes are considered in determining the displacement due to
seismic loading of the columns with foundation flexibility neglected. No effects due to
rotatory inertia of the supported mass are considered.
A flow diagram for the design of a bridge pier using this method is provided in Figure 6.
Nearly identical procedures are used for both the immediate operation and life safety
performance objectives. The differences are that the limiting curvature ductility value should
be specified at a lower value based on the desire to limit structural damage and a limit on
residual displacement must also be considered. The end result for both performance
objectives is a seismic base moment for flexural reinforcement design of the plastic hinge,
with the more critical value being used for design. The life safety design procedure is
presented first, followed by the immediate operation procedure, and recommended capacity
design considerations.
LIFE SAFETY PERFORMANCE OBJECTIVE
Prior to performing the seismic design, a preliminary column size must be selected based
on gravity load and anticipated seismic demands. As a first step, an appropriate damage state
must be selected along with an appropriate curvature ductility value. For columns that are
appropriately detailed for significant ductility demand, it is likely that curvature ductility
levels of 14-18 can be achieved. Yield and plastic displacement can subsequently be
determined from Equations 7 and 9, respectively. The resulting displacement ductility
capacity is:
y DS pu
y y
ημΔ
Δ + ΔΔ= =Δ Δ
(12)
To determine the actual inelastic displacement ratio, CR, an initial guess of the structural
period is required. With the initial guess for period, the inelastic displacement ratio can be
determined from Equation 2 and the generalized modification factor from Equation 3. A new
structural period is then calculated by using the relationship from Equation 5 using the
updated modification factor. This process is repeated until the structural period has
converged. The seismic base shear coefficient is then calculated based on principles of
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Period, T
Constant displacement
Constant velocity
Constant acceleration
Tα ⋅
Peak ground displacement
Figure 1. Assumed displacement design spectrum
Period, T
90th PercentileMedian
0.5
0.3
1 11.7RC
TμΔ −
= +⋅
Dispersion decreases
CR
1
Lognormal Fit
Figure 2. Variation of inelastic displacement ratio compared to period
h h
yφ yφpφ
uφ
Idealized Behavior
Actual Behavior
Idealized Behavior
Actual Behavior
pl
Figure 3. Curvature profile a.) At idealized yield and b.) At ultimate
JOSE RESTREPO
Note
the actual max curvature equals phi_u as in the equivalent p.h. - correct this figure
Curvature
Mom
ent
Damage State I
Damage State II
Damage State III
' 0.0c g
Nf A
=
' 0.3c g
Nf A
=
Figure 4. Example moment-curvature results with three example damage states
Figure 5. Cantilever bridge pier a.) Transverse elevation and b.) Structural model
LS
yy D
λ εφ
⋅=
( )1p yφφ μ φ= − ⋅
Assume T
Yes
No
Use
Cal
cula
ted
T
ImmediateOccupancy
No
2p
p p p
ll hφ
⎛ ⎞Δ = ⋅ ⋅ −⎜ ⎟
⎝ ⎠
TConverge
2
3y
y
hφ ⋅Δ =
22 ysC
T gπ Δ⋅⎛ ⎞= ⋅⎜ ⎟
⎝ ⎠
design sM C W h= ⋅ ⋅
2 2( 1) ( 1) 1Q RC C CΔ = − + − +
uTC αΔ
Δ=
⋅
y DS pu
y y
ημΔ
Δ + ΔΔ= =Δ Δ
Modify a for 2% damping
( )11R uκμ −
ΔΔ = Δ −
,maxR RΔ > Δ
Yes
Modify to satisfy requirement
pΔ
Yes
y DS pu
y y
ημΔ
Δ + ΔΔ= =Δ Δ
No
DBD
Bridge geometry, materials,and seismic hazard
Damage limit states, φμIO
0.5
0.3
1 11.7RC
TμΔ −
= +⋅
ImmediateOccupancy
Stop
No
Rep
eat P
roce
dure
For
Life
Saf
ety
Yes
Figure 6. Design flow chart
0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 250
Slope, α, Characterizing a Portion of the Displacement Spectrum [mm/sec]
Bas
e Sh
ear C
oeff
icie
ntD = 1.8 mh = 6.0 m
0
Low SeismicModerate Seismic
High Seismic
Desirable design coefficient
Unfeasib
le Domain
Figure 7. Influence of variables α and φμ on the design for the life-safety performance objective
0
0.01
0.02
0.03
0.04
0.05
0 10 20 30 40 50
Slope, α, Characterizing a Portion of the Displacement Spectrum [mm/sec]
Bas
e Sh
ear C
oeff
icie
nt D = 1.8 mh = 6.0 mk = 0
6φμ =
Figure 8. Influence of plastic drift ratio on immediate occupancy design procedure
0
0.05
0.1
0.15
0.2
0.9 1.2 1.5 1.8 2.1 2.4
Column Diameter [m]
Bas
e Sh
ear C
oeff
icie
nt
h = 6.0 ma = 50 mm/seck = 0
1.5%Rθ =2.0%Rθ =
6φμ =
0
0.05
0.1
0.15
0.2
2 4 6 8 10
Column Height [m]
Bas
e Sh
ear C
oeff
icie
nt D = 1.2 ma = 50 mm/seck = 0
6φμ =
Figure 9. a.) Influence of column diameter and b.) Influence of column height on immediate occupancy design procedure considering various plastic drift ratios
00.10.20.30.40.50.60.70.80.9
1
1.2 1.4 1.6 1.8 2 2.2 2.4
Column Diameter [m]
Bas
e Sh
ear C
oeff
icie
nt a = 200 mm/secmf = 18
8 mh =9 mh =
Figure 10. Influence of column diameter and height on life safety design procedure
Damage State
Probability of Exceedance Performance Level
I 50% in 50 years Immediate Occupancy II 10% in 50 years Not Considered III 2% in 50 years Life Safety
Table 1. Comparison between damage states and seismic demand