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, mjtobolski@ucsd.edu b) Associate Professor, University of California San Diego, Department of Structural Engineering, 9500 Gilman Dr. MC 0085, La Jolla, CA 92093-0085, jrestrepo@ucsd.edu

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

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.

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

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

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

structural dynamics assuming elastic-perfectly plastic behavior:

22 y

sCT gπ Δ⋅⎛ ⎞= ⋅⎜ ⎟

⎝ ⎠ (13)

where g is the acceleration due to gravity. This base shear coefficient is then be used to

determine a flexural design base moment:

design sM C W h= ⋅ ⋅ (14)

This design moment will later be compared with the design moment for the immediate

operation performance objective to determine the governing performance objective for

flexural design.

IMMEDIATE OPERATION PERFORMANCE OBJECTIVE

The curvature ductility selected for the immediate operation performance objective

should be based on strain damage limit states in the column representative of minor and

repairable damage, such as Damage State I as previously described. It is expected that the

curvature ductility based on strain states will be between 4-8 for this performance objective.

An appropriate residual drift ratio should also be considered. Based on Japanese

recommendations following the Kobe Earthquake, a reasonable maximum value for residual

drift for immediate use following a seismic event is 1% (Japan Road Association, 2002). As

stated previously, lower levels of damping should be considered for this performance

objective. The authors recommend considering mass proportional damping of 2%

representing the lower level of damage that is expected for this level of seismic demand.

In specifying lower curvature ductility values it is important to consider the relationship

between curvature ductility and the spread of plasticity in reinforced concrete columns. Prior

research has shown that a non-linear variation of plastic hinge length exists with varying

levels of curvature ductility (Hines et al., 2004). Significant non-linearity between curvature

ductility and plastic hinge length typically stabilizes around curvature ductility of 5. At

ductility levels that are larger than 5, the plastic hinge length continues to grow due in part to

strain hardening of the reinforcing steel, but this increase is much less significant than that

below ductility of 5 and can be idealized as constant. The authors recommend using a plastic

hinge length the same as that recommended for the life safety performance objective when

curvature ductility is greater than 5 and a linear variation from zero to this value for curvature

ductilities between 1 and 5.

As a goal of the immediate operation condition is that the structure have minimal damage

and only minor residual displacements, both criterion must be considered. In order to

determine which case controls, the residual displacement should be calculated based on the

damage criteria and compared to the residual drift limit specified. If this residual

displacement is greater than the specified limit, then the ultimate displacement must be

reduced to satisfy residual displacement requirements.

Unfortunately, appropriate methods to determine residual displacement from a design

aspect have yet to be fully developed (Ruiz-Garcia and Miranda, 2005; Pampanin et al.,

2002; Kawashima et al., 1998) especially considering soil-foundation-structure interaction.

Consequently, the authors recommend considering the residual displacement in a

deterministic basis equal to that obtained upon unloading from peak displacement. In typical

reinforced concrete systems, unloading softening is observed resulting in residual

displacements that are less than the plastic displacement. This behavior can be considered

through the use of an unloading rule that considers Emori Unloading (Otani, 1974). For the

purposes of this study, the residual displacement is defined as:

( )11R uκμ −

ΔΔ = Δ − (15)

where κ is a factor between 0 and 0.5 that is used to consider Emori unloading. It is apparent

from Equation 15 that the residual displacement is a function of the ultimate displacement,

which is one of the results from this design procedure. As the Emori unloading factor is an

exponential multiplier and can vary anywhere between 0 and 0.5, there is no general closed

form solution to determine the ultimate displacement based on a given residual displacement

limit. However, for the extreme values where κ is taken as 0 and 0.5, the plastic

displacement limit based on residual offset considerations is:

,max0 Rp

DS

κ

η= Δ

Δ = (16)

and,

2

,max0.5,max

412

y y R yp R

DS

κ

η=

⎡ ⎤Δ + Δ Δ −Δ⎢ ⎥Δ = Δ +⎢ ⎥⎣ ⎦

(17)

where ,maxRΔ is the specified maximum residual displacement. For the general case where the

alpha value is not 0 or 0.5, a simple method to determine the maximum plastic displacement

based on residual offset considerations is through iteration, similar to what is done to

determine RC . Alternatively, an approximation for the maximum plastic displacement limit

based on residual offset considerations for an arbitrary κ is a linear variation between the

two extreme values:

( ) ( )0 0.5 0 2,max .max

1( ) 2 4p p p p R y y R yDS

κ κ κκ κ κη

= = = ⎡ ⎤Δ = Δ + Δ −Δ = Δ + Δ + Δ Δ −Δ⎢ ⎥⎣ ⎦ (18)

For the purposes of the parametric studies in this paper, an Emori unloading factor of 0 will

be considered.

It is important to note that in situations where residual drift considerations limit the

immediate operation design, it is possible that the curvature ductility will be lower than 5 and

the plastic hinge length would not have stabilized. In this case, iteration must occur to

account for the effect of decreasing plastic hinge length.

Once the plastic displacement has been determined, the remainder of the procedure is

identical to that for life safety design.

MEMBER DESIGN

After carrying out the design method for both performance objectives, the largest design

moment shall be used for design of flexural reinforcement in the plastic hinge region. In the

event that the axial load varies due to seismic excitation, the design moment shall be

considered in conjunction with the minimum axial load to designing flexural reinforcement.

Material properties used for design in the plastic hinge region should be nominal strength

values. This design moment is used only for design of flexural reinforcement in the plastic

hinge region; capacity design must be used for the design of all other portions of the column

(Priestley et al., 1996; Caltrans, 2004).

For example, once the column flexural reinforcement has been selected, design shear

demand for the column should be determined based on capacity design principles. To

determine the overstrength shear demand, the nominal moment resistance can be increased

by a series of factors to account for all sources of overstrength or moment-curvature analysis

can be conducted using the actual reinforcement selected with expected material properties

(including work hardening) along with the largest axial load expected. It is essential to

consider the maximum shear that could be induced in the column from flexural hinging in

order to ensure the selected mode of inelastic deformation is able to be achieved and

maintained (Kowalsky et al., 1994).

Capacity design should also be used when bar curtailment is desired. In this case the

overstrength moment demand should also be considered to determine the appropriate location

to terminate flexural reinforcement. Along with overstrength, curtailment of flexural

reinforcement must also be performed considering tension shift effects (Park and Paulay,

1975). Capacity design procedures are essential to ensure the intended deformation mode and

seismic response is achieved.

PARAMETRIC ANALYSES

A variety of analyses were conducted in order to determine the influence that certain

parameters may have on design using the proposed displacement-based design method. The

influence of seismic demand, curvature ductility, residual drift limit, column height, and

column diameter were investigated. For all analyses conducted below, displacement

reduction factors were considered equal to unity.

SEISMIC DEMAND AND CURVATURE DUCTILITY

The effect of seismic demand and curvature ductility was investigated by performing the

design procedure for a range of values of α and φμ . The following parameters were held

constant: column diameter (1.8 m), column height (6.0 m), plastic hinge length (½ column

diameter), CQ (1.25), and yield strain of steel (0.00215). Results from these analyses are

shown in Figure 7.

To gain insight into the relationship between immediate operation and life safety

performance objective demands, consider a desired based shear coefficient equal to 0.1. Such

a flexural design based shear coefficient typically facilitates construction without significant

congestion of longitudinal reinforcing steel. If the column was detailed to achieve significant

ductility ( 22φμ = ) with the immediate operation ductility demand limited to 6φμ = , the ratio

between the relative seismic demand would be on the order of 2. This implies that the

frequently occurring earthquake is significant in relation to the maximum credible

earthquake.

In the same figure three regions are labeled: low seismic, moderate seismic and high

seismic. These regions relate to the required level of seismic detailing for the desired flexural

design base shear coefficient of 0.1. For the specified column configuration (height and

diameter) and desired design coefficient, the required level of seismic detailing can be readily

determined by finding in what region the slope,α , lies. For low seismic (i.e. 6φμ ≤ )

minimal prescriptive seismic detailing is required, for moderate seismic (i.e. 6 14φμ< ≤ )

some level of detailing is required and for the high seismic (i.e. 14 22φμ< ≤ ) significant

confinement and detailing is required. The region past 22φμ = is labeled an unfeasible

domain as reliably such curvature ductility is unlikely to be achievable.

RESIDUAL DRIFT RATIO

Figure 8 provides a comparison of the design base shear coefficient for various plastic

drift ratios, Rθ , and for a constant curvature ductility, 6φμ = , which is a value within the

range expected for the immediate operation limit state. These values were obtained using

identical parameters as the previous case with the addition of the Emori unloading factor,

0κ = . It is evident that the plastic drift ratio selected will have a significant impact on the

design for the immediate operation performance objective. From inspection of the design

equations, it is apparent that for a given column diameter and height, the controlling

immediate operation consideration can be readily determined based on curvature ductility

and residual drift limits when considering a structure that unloads with the same loading

stiffness. From the results for this specific situation, it is apparent that the curvature ductility

limit of 6 will result in similar design requirements as a residual drift limit of 1%. For cases

where the Emori unloading factor is taken greater than 0, the curvature limit would stand out

as the controlling criteria. For the purposes of a frequently occurring earthquake, it is

desirable to specify a residual drift ratio less than or equal to 1% to satisfy the immediate

operation performance objective. For the case shown, it appears that to meet the objectives of

this performance objective both the residual drift limitation and curvature criterion specified

will yield nearly identical design demands; however as the column diameter and height are

varied, this will no longer hold true.

COLUMN DIAMETER AND HEIGHT

The effect of column diameter and height on immediate operation performance objective

was studied by varying these two parameters for a given seismic demand (i.e. α set equal to

50 mm/sec). It is observed that for a given column height as the column diameter is increased

the curvature ductility limitation will become slightly more controlling as compared to the

residual drift limitation (Figure 9a). However, it is apparent from this figure that for a

constant curvature ductility there is not a significant increase in the flexural design base shear

coefficient as the column diameter increases. Similarly, when column height is increased

with the column diameter and seismic demand remaining constant, the curvature ductility

limit becomes slightly more controlling as compared to the residual drift limit (Figure 9b).

However, both of these variations are fairly trivial and allude to that fact that the criterion

that controls the immediate operation criteria is fairly insensitive to the column dimensions.

The influence of column diameter and height on the life safety performance objective is

shown in Figure 10. For the life safety performance objected, a similar trend is observed as

for the immediate operation case, where an increase in column diameter leads to an increase

in flexural design moment due to the decrease in plastic curvature for a given curvature

ductility due to the decrease in yield curvature. A counterintuitive trend is observed for the

taller columns where the base shear coefficient appears almost completely insensitive to the

column diameter. This would signify that for the taller column height, the column dimension

could be selected based on a desired reinforcement ratio allowing for freedom in design that

would facilitate constructability.

Also shown on this figure are certain column aspect ratios. The results of this design

procedure are not shown for aspect ratios less than 2.5 as such ratios are not practical for

design. A significant observation is that the base shear coefficient is directly related to the

column diameter and not the aspect ratio. As the column height is typically defined based on

site considerations, the trends observed signify that his displacement-based procedure

facilitates rapid manipulation of the column diameter as a means to optimize the flexural

design.

CONCLUSIONS

Displacement-based design procedures provide rational means to design structures to

resist seismic actions and can provide excellent means to predict structural damage. A two-

level displacement-based method was presented which allows for design considering both

immediate operation and life safety performance objectives. This method was developed in a

probabilistic context to allow for the explicit consideration of uncertainties associated with

seismic design. The goal of the immediate operation performance objective is to limit

structural damage and residual drifts, while the goal of the life safety performance objective

is to prevent loss of life. While iteration is required in carrying out this design method, the

procedure lends itself well to simple computer coding which can perform all necessary

iterations. Though this procedure has been presented for single degree-of-freedom systems

subjected to far field ground motions, it has the potential to be extended to multiple degree-

of-freedom systems and near fault systems through the development of appropriate inelastic

displacement ratios and multi-modal behavior for these situations. In order to ensure the

desired structural mode of response is achieved, capacity design considering all sources of

overstrength must be performed to ensure premature failure does not occur.

A parametric study was performed in order to determine the influence that certain design

variables have on the seismic demand calculated with the method presented. Results from

this study indicate that the proposed method is in line with trends observed in the seismic

behavior of structures. An overview of the results of this parametric analysis is:

• Based on curvature ductility limits, for common seismic demand levels life safety

performance objectives will control the flexural design as compared to immediate

operation consideration.

• Through strict residual drift limits, the immediate operation limit state has the potential

to control the design. However, the resulting curvature ductility demand for immediate

operation indicates the structure would likely not exhibit appreciable damage.

• For immediate operation design at a given seismic demand level, an increase in

column diameter causes curvature ductility to become marginally more controlling as

compared to residual drift.

• Increasing column height results in curvature ductility limits becoming slightly more

controlling for immediate operation design; however this increase appears negligible.

• Flexural base shear coefficients for life safety increase rather linearly with increasing

column diameter; however this trend appears minor for taller columns signifying the

flexural design base shear coefficient is insensitive to column diameter for tall

columns.

• A decrease in column height can lead to a significant increase in the design base shear.

Further work is needed in order to extend this design procedure to multiple degree-of-

freedom systems such as multi-column bents. Near fault effects must also be considered

through further investigation of their effects on inelastic displacement demands. Instrumental

to these two tasks is the development of inelastic displacement relationships for columns in

double curvature and near fault sites.

The proposed procedure allows designers to design aerial structures rationally and

adequately consider multiple design level events to meet the demands of emerging

performance-based seismic design methodologies.

ACKNOWLEDGEMENTS

TEXT

APPENDIX A – DESIGN EXAMPLE

The following provides a design example using the displacement-based design procedure

presented in this paper. A variety of modification factors are used in this example, however

these values are not considered appropriate for all design situations.

PROBLEM STATEMENT

A portion of an elevated highway structure, similar to the one shown in Figure 5, shall be

designed to resist seismically induced displacements based on the design procedure presented

in this paper. The column is 7.2 meters tall from its base to the center of mass of the

superstructure. Preliminary sizing and form availability indicates an appropriate column

diameter is 1.8 meters (aspect ratio of 4). The equivalent plastic hinge length can be assumed

to equal one-half the column diameter. Seismic demands for the life safety performance

objective can be characterized by a slope of the 5% damped elastic displacement design

spectrum equal to 165 mm/sec. For the immediate operation performance objective, the slope

of the 2% damped elastic displacement design spectrum is 50 mm/sec. The displacement

modification factor, CQ, can be taken as 1.25.

Moment-curvature analyses have indicated that a curvature ductility of 18 may be used

for life safety performance level, and 6 for immediate operation performance level. The

acceptable residual drift limit for the immediate operation performance objective is 1%. The

plastic displacement reduction factor for immediate operation shall be taken as 0.90 while for

the life safety a factor of 0.85 should be used.

The nominal concrete compressive strength shall be taken as 30 MPa with the probable

concrete strength equal to 45 MPa. The nominal yield strength of the reinforcing steel is 414

MPa while the probable yield strength is 460 MPa with the modulus of elasticity of 200 GPa.

The seismic weight of system is 8000 kN (axial load ratio of 0.10), but due to seismic loading

the axial load on the column may vary from 6500 kN to 9500 kN.

Design the reinforced concrete column for flexure based on the displacement-based

design procedure presented and determine the design shear force based on capacity design.

SOLUTION

To determine the seismic design moment, perform the design procedure for both

performance objectives and determine the controlling case.

Immediate Operation Performance Objective

The idealized yield curvature, per Equation 6, is:

62.25 0.00207 rad2.59 10 mm1800mmy

y Dλ ε

φ −⋅ ⋅= = = ⋅

The reference yield displacement, per Equation 7, is:

( )262 rad2.59 10 7200mmmm 44.7mm3 3

yy

hφ −⋅ ⋅⋅Δ = = =

The plastic curvature based on curvature ductility considerations, per Equation 8, is:

( ) 6 5rad rad1 (6 1) 2.59 10 1.29 10mm mmp yφφ μ φ − −= − ⋅ = − ⋅ ⋅ = ⋅

The resulting plastic displacement, per Equation 9, is:

5 900mmrad1.29 10 900mm 7200mm 78.6mmmm2 2p

p p p

ll hφ −⎛ ⎞ ⎛ ⎞Δ = ⋅ ⋅ − = ⋅ ⋅ ⋅ − =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

The limiting value of residual displacement is:

,max ,max 0.01 7200mm=72mmR R hθΔ = ⋅ = ⋅

From inspection it is apparent that the residual drift limitation will control this design,

though the curvature limit yields very similar results. The updated plastic curvature, based on

residual drift limit limitations is:

,max 572mm rad1.19 10 mm900mm900mm 7200mm22

Rp

pp

ll h

φ −Δ= = = ⋅

⎛ ⎞ ⎛ ⎞⋅ −⋅ − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

Determine the actual curvature ductility:

6 5

6

rad rad2.59 10 1.19 10mm mm 5.6rad2.59 10 mm

y p

yφ

φ φμ

φ

− −

−

⋅ + ⋅+= = =

⋅

Note: As the curvature ductility remains greater than 5, the assumption that the plastic

hinge length is one-half the column diameter remains valid.

The reliable ultimate displacement capacity, per Equation 11, is:

44.7mm 0.90 72mm 109.5mmu y DS pηΔ = Δ + Δ = + ⋅ =

Reliable ultimate displacement ductility, per Equation 12, is:

109.5mm 2.444.7mm

u

y

μΔ

Δ= = =Δ

As an initial guess say structural period, T, is 1 second. Inelastic displacement ratio, per

Equation 2, is:

0.5 0.5

0.3 0.3

1 2.4 11 1 1.331.7 1.7 1RC

TμΔ − −

= + = + =⋅ ⋅

Generalized displacement modification factor, per Equation 3, is:

2 2 2 2( 1) ( 1) 1 (1.25 1) (1.33 1) 1 1.42Q RC C CΔ = − + − + = − + − + =

Recalculated period, based on Equation 5, is:

109.5mm 1.63secmm1.42 60 sec

uTC αΔ

Δ= = =

⋅ ⋅

Summary of iteration to determine period, T:

T [sec] CR CΔ 1 1.33 1.42

1.63 1.29 1.38 1.67 1.28 1.38 1.67 1.28 1.38

Flexural design based shear coefficient, per Equation 13, is:

2 2

2

2 2 44.7mm 0.06mm1.68sec 9810 sec

ysC

T gπ πΔ⋅ ⋅⎛ ⎞ ⎛ ⎞= ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Flexural design moment for plastic hinge, per Equation 14, is:

, 0.06 8000kN 7.2m 3696kN mdesign IO sM C W h= ⋅ ⋅ = ⋅ ⋅ = ⋅

Life Safety Performance Objective

The plastic curvature, per Equation 8, is:

( ) 6 5rad rad1 (18 1) 2.59 10 4.40 10mm mmp yφφ μ φ − −= − ⋅ = − ⋅ ⋅ = ⋅

The resulting plastic displacement, per Equation 9, is:

5 900mmrad4.40 10 900mm 7200mm 267.2mmmm2 2p

p p p

ll hφ −⎛ ⎞ ⎛ ⎞Δ = ⋅ ⋅ − = ⋅ ⋅ ⋅ − =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

The reliable ultimate displacement capacity, per Equation 11, is:

44.7mm 0.85 267.2mm 271.9mmu y DS pηΔ = Δ + Δ = + ⋅ =

Reliable ultimate displacement ductility, per Equation 12, is:

271.9mm 6.144.7mm

u

y

μΔ

Δ= = =Δ

As an initial guess say structural period, T, is 1 second. Inelastic displacement ratio, per

Equation 2, is:

0.5 0.5

0.3 0.3

1 6.1 11 1 1.861.7 1.7 1RC

TμΔ − −

= + = + =⋅ ⋅

Generalized displacement modification factor, per Equation 3, is:

2 2 2 2( 1) ( 1) 1 (1.25 1) (1.86 1) 1 1.90Q RC C CΔ = − + − + = − + − + =

Recalculated period, based on Equation 5, is:

271.9mm 0.87secmm1.90 165 sec

uTC αΔ

Δ= = =

⋅ ⋅

Summary of iteration to determine period, T:

T [sec] CR CΔ 1 1.86 1.90

0.87 1.90 1.93 0.85 1.90 1.94 0.85 1.91 1.94

Flexural design based shear coefficient, per Equation 13, is:

2 2

2

2 2 44.7mm 0.25mm0.85sec 9810 sec

ysC

T gπ πΔ⋅ ⋅⎛ ⎞ ⎛ ⎞= ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Flexural design moment for plastic hinge, per Equation 14, is:

, 0.25 8000kN 7.2m 14359kN mdesign LS sM C W h= ⋅ ⋅ = ⋅ ⋅ = ⋅

COLUMN DESIGN

Based on the two performance objectives considered, the life safety criteria will control

the design of flexural reinforcement in the column. As the axial load in the column can vary

during a seismic event, consider the minimum axial force in combination with the design

moment from the life safety performance objective to determine the required flexural

reinforcement. Nominal material properties should be considered in performing this design.

Making assumptions for the clear cover and transverse reinforcement, a reinforcing ratio of

approximately 1.7% will be adequate to resist to seismic demands.

Following the selection of the actual size and number of reinforcing bars, the section was

modeled in a moment-curvature program using expected material properties considering

work hardening along with the largest axial load expected. The ultimate moment determined

from this analysis was 1.6 times greater than the moment used for flexural reinforcement

design. Consequently, the shear reinforcement in the column should be design based on a

shear force equal to:

, 1.6 14359kN m 3091kN7.2m

o design LSdesign

MV

hΩ ⋅ ⋅ ⋅

= = =

where Ω is the overstrength factor.

As a comparison, the design base moment using a force-based approach considering the

effective column stiffness equal to half the gross stiffness yields a design moment of 49,738

kN-m, or about 3.5 times the design moment obtained through this procedure.

REFERENCES

AASHTO. 2004. LRFD Bridge Design Specifications. American Association of State Highway and

Transportation Officials, Washington, D.C.

Abrahamson, N.A. and Bommer, J.J. 2005. Probability and Uncertainty in Seismic Hazard Analysis.

Earthquake Spectra. Vol. 21, No. 2. pp. 603-607.

Aschheim, M.A. and Black, E.F. 2000. Yield Point Spectra for Seismic Design and Rehabilitation.

Earthquake Spectra. Vol. 16, No. 2. pp. 317-336.

Browning, J.P. 2001. Proportioning of Earthquake-Resistant RC Building Structures. American

Society of Civil Engineers Journal of Structural Engineering. Vol. 127, No. 2. pp. 145-151.

Caltrans. 1999. Memo to Designers 20-1. California Department of Transportation, Sacramento, CA.

Caltrans. 2004. Seismic Design Criteria Version 1.3. California Department of Transportation,

Sacramento, CA.

Calvi, G.M. and Kingsley, G.R. 1995. Displacement-Based Seismic Design of Multi-Degree-of-

Freedom Bridge Structures. Earthquake Engineering and Structural Dynamics. Vol. 24, No. 9.

pp. 1247-1266.

CEN. 1998. Eurocode 8 Design Provisions for Earthquake Resistance of Structures, European

Prestandard ENV Comité Européen de Normalisation, Brussels.

Chopra, A.K. 2001. Dynamics of Structures Theory and Applications to Earthquake Engineering, 2nd

Edition. Prentice Hall, New Jersey.

Chopra, A.K. and Goel, R.K. 2001. Direct Displacement-Based Design: Use of Inelastic vs. Elastic

Design Spectra. Earthquake Spectra. Vol. 17, No. 1 pp. 47-64.

Crowley, H., Bommer, J.J., Pinho, R. and Bird, J. 2005. The Impact of Epistemic Uncertainty on

Earthquake Loss Model. Earthquake Engineering and Structural Dynamics. Vol. 34. pp. 1653-

1685.

Faccioli, E., Paolucci, R. and Rey, J. 2003. Displacement Spectra for Long Periods. Earthquake

Spectra. Vol 20, No 2 pp. 347-376

FEMA. 1997. NEHRP Guidelines for the Seismic Rehabilitations of Buildings FEMA 273. Federal

Emergency Management Agency, Washington, D.C.

Fu, Y and Cherry, S. 1999. Simplified Seismic Code Design Procedure for Friction Damped Steel

Frames. Canadian Journal of Civil Engineering. Vol. 26 pp. 55-71.

Hines, E.M., Restrepo, J.I., and Seible, F. 2004. Force-Displacement Characterization of Well-

Confined Bridge Piers. American Concrete Institute Structural Journal. Vol. 101, No. 4 pp. 537-

548.

Imbsem, R.A. 2006. Recommended LRFD Guidelines for the Seismic Design of Highway Bridges.

Japan Road Association. 2002. Design Specification of Highway Bridges Part V: Seismic Design.

Japan.

Kawashima, K. and Aizawa, K. 1986. Modification of Earthquake Response Spectra with Respect to

Damping Ratio. Proceedings of the Third U.S. National Conference on Earthquake Engineering.

Vol. 2 pp. 1107-1116

Kawashima, K., MacRae, G.A., Hoshikuma, J., and Nagaya, K. 1998. Residual Displacement

Response Spectrum. American Society of Civil Engineers Journal of Structural Engineering. Vol.

124, No. 5, pp. 523-530.

Kowalsky, M., Priestley, M.J.N., and MacRae, G. 1994. Displacement-Based Design: A Methodology

for Seismic Design Applied to Single Degree of Freedom Reinforced Concrete Structures. SSRP-

94/16. University of California, San Diego. La Jolla, CA.

Miranda, E. and Bertero, V.V. 1994. Evaluation of Strength Reduction Factors for Earthquake-

Resistant Design. Earthquake Spectra. Vol. 10, No. 2. pp. 357-379.

Moehle, J.P. 1992. Displacement-Based Design of RC Structures Subjected to Earthquakes.

Earthquake Spectra. Vol. 8, No. 3. pp. 403-428.

NCHRP. 2001. Comprehensive Specification for the Seismic Design of Bridges Revised LRFD Design

Specifications, Third Draft. National Cooperative Highway Research Program Project 12-49.

Washington, D.C.

Otani, S. 1974. Inelastic Analysis of R/C Frame Structures. American Society of Civil Engineers

Journal of the Structural Division. Vol. 100, No. ST7. pp. 1433-1449.

Pampanin, S., Christopoulos C., Priestley, M.J.N. 2002. Residual Deformations in the Performance-

Seismic Assessment of Frame Structures. Research Report No. ROSE-2002/02. European School

for Advanced Studies in Reduction of Seismic Risk, Pavia, Italy.

Park, R. and Paulay, T. 1975. Reinforced Concrete Structures. John Wiley & Sons, Inc. New York.

Priestley, M.J.N. 2003. Myths and Fallacies in Earthquake Engineering, Revisited. IUSS Press, Pavia,

Italy.

Priestley, M.J.N. and Kowalsky, M.J. 2000. Direct Displacement-Based Seismic Design of Concrete

Buildings. Bulletin of the New Zealand National Society for Earthquake Engineering. Vol. 33,

No. 4, pp. 421-444.

Priestley, M.J.N., Seible, F., and Calvi, G.M. 1996. Seismic Design and Retrofit of Bridges. John

Wiley & Sons, Inc. New York.

Restrepo-Posada, J.I., Dodd, L.L., Park, R., and Cooke, N. 1994. Variables Affecting Cyclic Behavior

of Reinforcing Steel. Journal of Structural Engineering. Vol. 120, No. 11. pp. 3178-3196.

Rodriguez, M.E., Botero, J.C., and Villa, J. 1999. Cyclic Stress-Strain Behavior of Reinforcing Steel

Including Effect of Buckling. Journal of Structural Engineering. Vol. 125, No. 6 pp. 605-612.

Ruiz-Garcia, J. and Miranda, E. 2003. Inelastic Displacement Ratios for Evaluating Existing

Structures. Earthquake Engineering and Structural Dynamics. 32:1237-1258. DOI:

10.1002/eqe.271.

Ruiz-Garcia, J. and Miranda, E. 2005. Residual Displacement Ratios for Assessment of Existing

Structurs. Earthquake Engineering and Structural Dynamics. 35:315-336. DOI:10.1002/eqe. 523.

Structural Engineers Association of California (SEAOC). 1999. Recommended Lateral Force

Requirements and Commentary, 7th Edition. Sacramento, CA.

Suleiman, M.T., Sritharan, S., and White, D.J. 2006. Cyclic Lateral Load Response of Bridge

Column-Foundation-Soil Systems in Freezing Conditions. Journal of Structural Engineering.

Vol. 132, No. 11. pp. 1745-1754.

Sullivan, T.J., Calvi, G.M., Priestley, M.J.N., and Kowalsky, M.J. 2003. The Limitations and

Performances of Different Displacement Based Design Methods. Journal of Earthquake

Engineering. Vol. 7, Special Issue 1 pp. 201-241.

Veltsos, A.S. and Newmark, N.M. 1960. Effect of Inelastic Behavior on the Response of Simple

System to Earthquake Motion. Proceedings to the 2nd World Conference on Earthquake

Engineering. pp. 895-912.

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

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