EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng Struct. Dyn. 2002; 31:719–747 (DOI: 10.1002/eqe.150)

A displacement-based approach for the seismic design

of continuous concrete bridges

Mervyn J. Kowalsky∗;†
Department of Civil Engineering; North Carolina State University; Box 7908; Raleigh; NC 27695; U.S.A.

SUMMARY
A displacement-based design procedure for continuous concrete bridges is proposed which can be in-
corporated into a performance-based design philosophy. The procedure is applicable to multi-degree-
of-freedom bridges with :exible or rigid superstructures, and for varying degrees of abutment restraint.
The background and development of the design procedure is presented =rst, followed by a series of
examples and validation studies using dynamic inelastic time-history analysis. The procedure is ap-
plied to the transverse response of bridge structures; however, it is equally well-suited for longitudinal
response. The results indicate that the process is able to capture non-linear deformation patterns and
thus reasonably control damage. Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: displacement-based design; bridges; limit states

INTRODUCTION

The philosophy of performance-based earthquake engineering has been su@ciently devel-

oped to the point where its de=nition is well-established, namely, to design a structural system
to sustain a pre-de=ned level of damage under a pre-de=ned level of earthquake intensity. The
Structural Engineers Association of California [1] has de=ned the marriage of the structural
performance and earthquake intensity as a ‘Performance Level’, and a suite of performance
levels as a ‘Performance Objective’. For example, three common performance levels that have
been at the heart of earthquake engineering long before the term ‘Performance-Based’ gained
favour are as follows:
(1) During a minor earthquake, deformation should be su@ciently small such that the level
of damage that occurs does not require repair.

∗ Correspondence to: Mervyn J. Kowalsky; Department of Civil Engineering; North Carolina State University; Box
7908; Raleigh; NC 27695; U.S.A.
† E-mail: kowalsky@eos.ncsu.edu

Received 15 January 2001

Revised 31 August 2001
Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 31 August 2001
720 M. J. KOWALSKY

(2) During a moderate earthquake, deformations should be controlled such that the damage
that occurs is repairable.
(3) During a severe earthquake, the deformation capacity of the system must not be ex-
ceeded, thus ensuring the safety of the occupants of the structure.
Taken together, these three performance levels form a performance objective that is often
applied to the routine, or basic structure. As a result, it can be argued that performance-based
engineering is conceptually nothing new and that all of the most recent eMorts in the area are
simply aimed at formalizing what good engineers have always sought to accomplish. While
this may be the case, it is clear that traditional seismic design methods, which are force-
based in nature, generally leave little to be desired from the perspective of achieving this goal
of performance-based engineering. It is generally agreed that deformations are more critical
parameters for de=ning performance, and as a result it is argued that seismic design methods
should largely be based on them.
It is the purpose of this paper to illustrate a direct displacement-based design procedure for
the design of bridge structures that can be employed to achieve performance-based earthquake
engineering.

LAYOUT OF THE PAPER

In this paper, the direct displacement-based design approach, which was initially developed
for single columns [2] and later extended to structural wall and moment frame buildings [3],
will be reviewed. This is then followed by a discussion of how the process is applied to the
performance-based design of bridge structures. This is followed with some design examples
and validation studies that were conducted with dynamic inelastic time-history analysis.

BACKGROUND OF DIRECT DISPLACEMENT-BASED DESIGN

The fundamental goal of the direct displacement-based design approach is to obtain a

structure which will reach a predetermined displacement when the structure is subjected to
an earthquake consistent with the design level event. The procedure is de=ned as a response
spectrum-based approach which utilizes the substitute structure methodology developed by
Gulkan and Sozen [4] to model an inelastic system with equivalent elastic properties. Con-
sider Figure 1 which represents a force–displacement hysteretic response for a typical well-
con=ned concrete bridge pier. The hysteretic response shown in Figure 1 can be modelled
with an equivalent elastic secant stiMness to the maximum response point (dotted line marked
K eM ) in conjunction with an equivalent viscous damping value, eM . The equivalent viscous
damping consists of two components: one due to the hysteretic energy dissipation, hyst , and
the other due to viscous damping, vis . The two parameters of eMective stiMness and equivalent
viscous damping form the basis of the substitute structure [4]. Since the substitute structure is
elastic, its response to a particular earthquake, and hence the response of the real structure can
be determined from elastic response spectra for the appropriate equivalent viscous damping.
In the direct displacement-based design approach, a structure is designed based on its
behaviour at maximum response. Therefore, after general parameters such as column height

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 721

Figure 1. EMective stiMness and damping properties.

and superstructure mass are established, the =rst step is selection of the target maximum
displacement, O m . After this is established, column yield displacement, O y , is estimated using
simpli=ed relations between section depth and yield curvature. Column displacement ductility,
O , is then obtained as

Om
O = (1)
Oy

Displacement ductility can then be related to hysteretic damping, hyst , with an assump-
tion on the shape of the hysteretic loop. Applying Jacobsen’s approach [5] to the Takeda
degrading-stiMness-hysteretic response [6] results in the relation given as Equation (2a) be-
tween displacement ductility and hysteretic damping [2; 7], where r is the ratio of the post-
elastic structural response curve to that of the initial elastic structural response curve. A value
of 0.05 for r is typical. The equivalent viscous damping is then obtained by adding a viscous
damping component to Equation (2a). A value of 5% is used here and the resulting equation
is shown as Equation (2b).

√ √
1 − ((1 − r)= O ) − r O
hyst = (2a)

√ √
1 − ((1 − r)= O ) − r O
eM = 0:05 + (2b)


The next step requires a selection of the demand which for direct displacement-
based design is a displacement response spectra generated for various levels of viscous
damping such as that shown in Figure 2. The design displacement spectra is then entered
with the target maximum displacement, O m , and intersected with the appropriate response
curve given by the equivalent viscous damping value of Equation (2b). Reading down to the
horizontal axis provides the eMective period of the structure at maximum response, TeM , as
in Figure 3. Through consideration of a single-degree-of-freedom (SDOF) oscillator, the eMec-
tive period at maximum response is related to the eMective stiMness at maximum

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722 M. J. KOWALSKY

Figure 2. ATC-32 soil type C, Magnitude 8, 0:6g PGA displacement response spectra: (a) Lin-
earization of ATC-32 spectra for design, (b) ATC-32 Spectrum, generated spectrum, and linear
design spectrum for 20 per cent damping.

Figure 3. Obtaining eMective period for direct displacement-based design.

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 723

Figure 4. Design :owcharts: (a) Direct displacement-based design, (b) Force-based design.

response, K eM ,

42 M
K eM = (3)
TeM2

The eMective stiMness from Equation (3) corresponds to the dashed line in Figure 1.
Therefore, the maximum lateral force, F, is easily obtained by multiplying the eMective stiM-
ness by the target maximum displacement as in Equation (4). Once the lateral force is estab-
lished, the system can be designed. Figure 4(a) represents a :owchart of the design approach

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724 M. J. KOWALSKY

for SDOF systems.

F = K eM O m (4)

If this approach is compared to the traditional force-based design approach as summarized

in Figure 4(b), then a signi=cant diMerence is noted. In the force-based approach, design forces
are based on the elastic period and force-reduction factors, while in the direct displacement-
based design approach, the design force is obtained by selecting a desired performance level
(displacement) and considering the inelastic response of the system (hysteretic damping).
Through the use of dynamic inelastic time-history analysis with the computer program
Ruaumoko [8], the SDOF approach was shown to be reasonably accurate (error usually less
than 15 per cent) in terms of reaching the design level displacements under the applied
force [2]. Therefore, since the procedure appears to be able to specify displacements for a
particular earthquake level, damage can in turn be speci=ed for a speci=c earthquake level,
since damage correlates well with deformation quantities, thus meeting the objective of a
performance-based design procedure. In the force-based design approach, damage and hence
performance are very di@cult to quantify for a particular earthquake level, since forces are
poor indicators of damage potential and force-reduction factors, which are meant to imply
damage levels, are highly variable and often disagreed with one code body or another.

PERFORMANCE-BASED EARTHQUAKE ENGINEERING (PBEE) OF BRIDGES

USING A DIRECT DISPLACEMENT-BASED APPROACH

The displacement-based design approach for the design of multi-degree-of-freedom (MDOF)

bridge structures can be reduced to the following basic steps, each of which will be explained
in detail in the subsequent sections.
(1) Selection of performance levels.
(2) Evaluation of target displacement pattern.
(3) Evaluation of an equivalent SDOF system.
(4) Application of the SDOF displacement-based design approach.
(5) Member design.

Selection of performance levels

Selection of the performance levels consists of identifying acceptable damage levels for a
suite of earthquake intensities. The number of earthquake levels considered is the choice of
the engineer, and the selection of acceptable damage is a function of bridge importance as well
as various economic and societal aspects. The engineer must determine the acceptable damage
to the bridge as a whole in a qualitative sense, which is then expressed quantitatively at the
member level. Acceptable damage at the member level is often de=ned as limit states relevant
to cracking, =rst yield, serviceability, damage control, and survival. Earthquake intensities are
often described in terms of return periods such as 50-year, 475-year, and 2500-year events.

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 725

Evaluation of target displacement pattern

Complexity in this design process, although minimal, is at its greatest in this portion of the
process. Unlike a single bridge column, a bridge that is composed of many columns that are
connected by a superstructure of de=ned :exibility will deform in a manner that is in:uenced
by the variations in strength, stiMness and mass distribution.
The =rst step in the process of evaluating the target displacement pattern is the calculation
of what is termed the ‘damage-based’ displacements for each column. It is assumed that
inelastic action will be con=ned to the columns of the bridge, and as a result, it is essential
to identify the lateral column displacement that will cause each individual column to reach
the target damage level. Even though the =nal displacement pattern is likely to consist of
only one column reaching its damage-based displacement, it is important to determine the
damage-based displacements of each column such that the critical column can be readily
identi=ed. In most cases, the shortest column will govern the selection of the displacement
pattern; however, depending on the modal response of the system, it is possible that a column
other than the shortest one will be the controlling member.
In this paper, the damage-based lateral displacements are based on a set of limiting longi-
tudinal strains that are consistent with the desired damage level. For example, in the case of
damage de=ned by the serviceability limit state, a concrete strain of 0.004 that represents the
onset of crushing may be selected. Similarly, a peak steel tension strain of 0.015 or residual
tension strain of 0.005 may be selected such that residual crack widths are su@ciently small,
thus not requiring repair.
The correlation between qualitative descriptions of damage and their quantitative counter-
parts expressed as strains is beyond the scope of this paper as many variables will enter
into that relationship, particularly at the damage control and survival levels. However, once
those strain levels are selected, evaluation of damage-based displacements, OD , follows from
the =rst principles. For example, using the plastic hinge method for member deformations,
the damage-based displacement based on concrete compression strain, c , and steel tension
strain, s , for a column is given by Equations (5) and (6), respectively. Note that each of
these requires knowledge of the section-neutral axis depth which is typically not known at
this stage of the design process.

 y L2eM
c
OD = − y L p Lclear + (5)
c 3
 
s y L2eM
OD = − y L p Lclear + (6)
D
− c 3

Recent research [9] has resulted in expressions for dimensionless limit-state curvatures,
LS , that are a function of axial load ratio and longitudinal steel ratio. A set of graphs
for diMerent steel ratios are shown in Figure 5 [9]. From these =gures, once a target steel
tension and concrete compression strain are established, the corresponding curvatures can
be easily obtained and the smaller of the two curvatures utilized to calculate the damage-
based displacements using Equation (7). Note that Equation (7) does not include a
component for shear deformation. In the case where shear deformation is signi=cant,
Equation (7) can be modi=ed by providing an additional term to account for shear

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726 M. J. KOWALSKY

Figure 5. Dimensionless curvatures for circular RC columns: (a) 1 per cent longitudinal
steel ratio, (b) 2 per cent longitudinal steel ratio, (c) 3 per cent longitudinal steel ratio,
(d) 4 per cent longitudinal steel ratio.

deformation.
y L2eM
OD = (LS − y )L p Lclear + (7)
3
In Equations (5)–(7), Lclear represents the clear column length, L p [10] represents the
column-plastic hinge length given by the greater values of Equations (8), where fy is the
longitudinal bar yield stress and db‘ is the longitudinal bar diameter. The yield curvature,
y , is given by Equation (9) for circular columns and Equation (10) for rectangular columns
[10], where y is the longitudinal bar yield strain, and D is the column diameter. The eMective
column length for elastic displacement calculations, LeM , is given in Equation (12).

L p = 0:08Lclear + 0:022fy db‘ (MPa) (8a)

L p = 0:044fy db‘ (MPa) (8b)

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 727

2:45 y
y = (9)
D
2:14 y
y = (10)
D
db‘
D
= D − Cover − (11)
2
LeM = Lclear + 0:022fy db‘ (MPa) (12)

Once the damage-based displacements for each column are obtained, an assessment of
the superstructure stiMness is required. The target displacement pattern for the bridge as a
whole will largely be a function of the rigidity of the superstructure. In the case where the
superstructure can be assumed to be rigid, the calculation of the displacement pattern for
a continuous bridge is straightforward as all points along the deck translate into an equal
amount. In the event of a strength and stiMness eccentricity due to a shift in the centre of
strength and stiMness from the centre of mass, which occurs when an asymmetric pattern of
column heights is utilized, the displacement pattern consists of a translational and rotational
component.
In the event that the superstructure is :exible, the displacements of each bent will vary
in accordance with the modal response of the system. In the unrealistic case of an in=nitely
:exible superstructure, each column is free to move on its own as an SDOF system.
Described =rst is the more general but more time-consuming process for evaluating the
deformation pattern when the superstructure is :exible. This will then be followed by the
simpler method that can be employed for bridges with rigid superstructures.

Flexible superstructures. Recall that in the direct displacement-based design procedure for
SDOF systems, design was performed by specifying a single target displacement—either ar-
bitrarily or from the consideration of a strain-based limit state as previously discussed. The
deformation of the column is assumed to be controlled by a =rst-mode response, which is
clearly an acceptable assumption for a cantilever column with a concentrated mass. However,
when extending the procedure to the design of entire bridge structures, the columns do not
deform independent of each other. The displacements of the columns must consider the con-
nection provided by the superstructure, and the modal response of the system, which is a
function of the relative column to superstructure stiMness.
To that end, the individual columns cannot be forced into a displacement pattern that is
incompatible with the mode shapes. In the case of a bridge with a short column in the centre
and tall columns on either side (Figure 6(b)), it is not reasonable to expect that all of the
columns will likely reach a displacement coinciding with a concrete crushing limit state. In
that case, only the central column will reach the speci=ed limit state, while the displacement
of the other columns will be a function of the displacement of the central column and the
mode shapes.
For a system which responds in an elastic manner, it is apparent that the mode shapes
should be based on elastic properties. Similarly, if the relative column and abutment stiMness
remain the same throughout the response, then the displaced shape is adequately de=ned by
elastic properties. However, if columns and abutments reach signi=cantly diMerent levels of

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728 M. J. KOWALSKY

Figure 6. Multi-span bridge con=gurations: (a) Symmetric, (b) Asymmetric.

inelastic response, then a displaced shape obtained from elastic properties will likely yield
erroneous results. In order to describe the modal response of an inelastic system such that a
displaced shape can be established, the concept of an e-ective mode shape is proposed. In
order to calculate the eMective mode shapes of a particular bridge structure, the secant stiMness
of the columns and abutments is utilized in the analytical eigenvalue problem.
What follows is a step-by-step approach for obtaining the target-displaced shape of a multi-
span bridge utilizing the eMective mode shapes in conjunction with the column strain criteria
and abutment displacement criteria. The objective is to obtain a displaced shape, whereby at
least one column or abutment reaches its desired damage level. The displacements of the other
columns and abutments are then obtained with reference to the eMective modal response.
Step 1: Evaluate mode shapes. Using either computer methods or hand calculations to
solve the eigenvalue problem, the mode shapes based on the column and abutment secant
stiMness values at maximum response are obtained. Elastic properties should be used for the
superstructure. For the =rst iteration of the procedure, the secant stiMness properties are not
known, therefore, it is suggested that a stiMness equal to 10 per cent of the uncracked section
stiMness be applied to columns expected to exceed their yield displacement, while a value of
50 per cent of the uncracked section stiMness is suggested for columns that are not expected
to exceed their yield displacement. Abutment stiMness for the =rst iteration can be assumed
to be 30 per cent of the initial elastic stiMness. It is important to recognize that exact stiMness
values are not required to start the procedure, and that any values will do initially. However,
the closer the estimates are to the actual values, the quicker the procedure will converge.
Step 2: Evaluate modal participation factors. The modal participation factors can be eval-
uated as follows:
i Mr
Pi = (13)
Ti Mi
where M represents a diagonal mass matrix and r is a unit vector.
Step 3: Evaluate likely displaced shape. The likely displaced shape is a function of several
modes, especially for the irregular bridge shown in Figure 6(b). The displaced shape for each
mode is obtained by Equation (14), where the index ‘i’ represents the bent number and the

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 729

index ‘j’ represents the mode number. Sd j is the spectral displacement for mode j; Pj is the
participation factor for mode j, and i; j is the modal factor for bent ‘i’ and mode ‘j’. O i; j is
then the displacement quantity associated with bent i and mode j.
Oi; j = i; j Pj Sd j (14)
In Equation (14), the spectral displacement, Sd j , for mode j is obtained by considering
each of the modal periods and system damping value. The modal periods are easily obtained
from the modal analysis previously performed.
The overall displaced shape is then obtained by an appropriate combination of modes such
as square root of the sum of the squares (SRSS) or complete quadratic combination (CQC).

Oi = O2i; j (15)
j

Utilizing the target displacements from the strain criteria, the target displacement pattern is
easily obtained. Only one displacement pattern will be compatible with the shape of Equa-
tion (15) and the strain-based displacement criteria. By scaling the displacement pattern of
Equation (15) to the strain-based displacements of each bent, the critical displacement pattern
is identi=ed as the one where the strain-based displacement targets are not exceeded.

Rigid superstructure simpli.cation. Although truly rigid superstructures are not likely to exist,
it is likely to be an acceptable approximation in many cases with the exception of very long
span systems. The routine multi-span box-girder-type structure can usually be assumed to have
a superstructure that is rigid in the transverse direction. Making such an assumption greatly
simpli=es the calculation of the target displacement pattern.
If the bridge superstructure is rigid in the transverse direction, then the displacement pat-
tern will indicate that all columns translate by the same displacement. In the case of irreg-
ular systems, this translation may be accompanied by a rotational component as shown in
Figure 7.
The target displacement pattern is then obtained as follows: (1) Evaluate the strain-based
displacements for each column. (2) Determine if the bridge is symmetric. If so, then the
target displacement pattern is a straight line drawn parallel to the bridge at a distance equal
to the smallest of the strain-based displacements as shown in Figure 7(a). If the bridge is not
symmetric, then the pattern will consist of a straight line rotated by an angle  as shown in
Figure 7(b).

Evaluation of an equivalent SDOF system

Having established the approach for determining the target-displaced shape, the next step in the
procedure is the characterization of the equivalent SDOF system. In order to characterize the
MDOF system (multi-span bridge) as an equivalent SDOF system, various system properties
must be de=ned. These are the (1) system target displacement, (2) system hysteretic damping
and (3) system mass.

System target displacement. In order to obtain a system displacement, work by Calvi et al.
[11], which is based on the initial studies by Biggs [12], is adopted. In this approach, the

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730 M. J. KOWALSKY

Figure 7. Possible displacement patterns for bridges without abutment restraint: (a) rigid
superstructure, plan view, (b) rigid superstructure with rotation, plan view, (c) :exible
superstructure, plan view.

system displacement, O sys , is de=ned by the requirement that the work done by the equivalent
SDOF system must be the same as that done by the MDOF system. The resulting expression
is

mi  2
O sys =  i (16)
mi i

In the case where the superstructure is assumed to be rigid and regular, the target system
displacement reduces to the lateral displacement of the superstructure.

System hysteretic damping. Hysteretic damping of individual column members is easily ob-
tained by utilizing relations between column ductility and hysteretic damping such as that
shown in Equation (2) [2] for the Takeda degrading-stiMness-hysteretic response [6]. The
ductility for each column member is obtained by dividing the displacement from the target
pro=le by the respective yield displacements.
Once the individual member damping values are obtained, they must be combined into
a value for the equivalent SDOF system. The approach suggested here weighs the damp-
ing in proportion to the work done by the member. The weighting factor, Q, is shown in

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 731

Equation (17). System damping is then obtained by Equation (18).

Qi = Fi O i (17)
 
 Q
sys =  i i (18)
i Qi

Calculation of the weighting factors in Equation (17) obviously requires the knowledge of
member forces. At this point in the procedure, member forces are not known; however, as
will be shown later (Equation (32)), forces are distributed among columns of equal diameter
in proportion to their secant stiMness to maximum response, which simpli=es to the inverse
of the column length. The weighting factor, Q, then reduces to the following equation, which
represents the column drift ratio:
Oi
Qi = (19)
Li
It is noted that the use of Equation (19) assumes that the abutments do not contribute to
the equivalent viscous damping of the system. If such contributions are to be included, then
Equation (17) should be utilized. As a result, the design procedure will then become iterative
as the knowledge of member forces will be required, and they are not obtained until the
end of the process. For the =rst iteration, it is suggested that a straight average damping as
in Equation (20) be employed, and that in the subsequent iterations, Equation (18) with the
weighting factors of Equation (17) be employed.

i
sys = i (20)
i
It is noted that Shibata and Sozen [13] previously suggested that hysteretic damping can be
weighted according to :exural strain energy for the substitute structure approach. In this paper,
the weighting factors are expressed as a work (Equation (17)) which allows us to directly
consider the abutment contribution, which would be more di@cult to quantify in terms of
:exural strain energy.

E-ective system mass. In order to complete the characterization of the equivalent SDOF
system, an eMective system mass must be identi=ed. Previously [14], the system mass was
taken to be the total inertia mass. However, this is likely to be incorrect. Instead, the system
mass should be that which requires work equivalence between the MDOF and SDOF systems
[11; 12]. The system mass is then obtained by Equation (21) where O i is the target displace-
ment for column i and m i is the mass associated with column i. EMective mass de=ned in this
manner considers the multiple modes of response since the displaced shape was established
with reference to multiple modes of response. If a structure responds in predominantly the
=rst mode, then the following equation will be approximately equal to the =rst modal mass:
  Oi 
MeM = mi (21)
O sys

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APPLICATION OF THE SDOF DISPLACEMENT-BASED DESIGN APPROACH

Now that an equivalent SDOF system has been identi=ed, the eMective period of the substitute
structure is obtained by entering the displacement response spectra with the target system
displacement and reading across to the appropriate response curve and down as was shown
in Figure 2. The eMective stiMness at maximum response of the equivalent SDOF system is
then obtained with Equation (22).
42 MeM
K eM ≈ (22)
TeM2
Utilizing the eMective stiMness and the target system displacement, the base shear force for
the equivalent SDOF system is then obtained with Equation (23).
Fsys = K eM O sys (23)

Member design

Having established the total design force, return to the MDOF system and determine the
column design forces. The total design force of Equation (23) is then the required resistance
for the entire MDOF structure.

Force distribution. As a matter of practicality, all columns should have the same longitudinal
steel ratio and column diameter. Such a requirement results in a simple calculation of the
column design forces. If all columns achieve a displacement ductility of at least one, then the
secant stiMness to maximum response of the individual columns, K eM , can be expressed with
Equation (24) assuming that the post-elastic slope of the force–displacement response is zero.
In Equation (24), K cr is the secant stiMness of the cracked section at the yield displacement
of the member, and O the displacement ductility. K cr can be expressed as in Equation (25),
where L is the column length, E is the elastic modulus of reinforced concrete, and I cr is
the moment of inertia of the cracked section at yield. EI cr is replaced by the constant X1
since all columns are of equal diameter and material. The displacement ductility is given by
Equation (26) where O m represents the maximum column displacement and O y represents the
column yield displacement. Substituting Equations (26) and (25) into Equation (24) results
in Equation (27).

K cr
K eM = (24)
O
3EI cr X1
K cr = = 3 (25)
L3 L
Om
O = (26)
Oy

X1 O y
K eM = (27)
L3 O m
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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 733

The yield displacement for a cantilever column can be expressed as Equation (28), where
X2 is a constant. For columns in double bending, the denominator in Equation (28) is 6
instead of 3; however, this has no eMect on the end result, unless the bridge contains columns
that are single and double cantilevers. Substituting Equation (28) into Equation (27) results
in Equation (29).

y L2
Oy = = X2 L2 (28)
3
X1 X2 L2
K eM = (29)
L3 O m

The column shear force is then given by Equation (30), which results in Equation (31)
after substitution of the Equation (29).

F = K eM O m (30)

X1 X2 L2 O m X1 X2
F= = (31)
L3 O m L

Equation (31) is then the lateral force resisted by a column. After dropping the constants,
the lateral force in each column, Fi , is then obtained with Equation (32), where Fc is the total
lateral force resisted by all the columns.
1=Li
Fi =  Fc (32)
1=Li
In the case where some of the columns remain elastic, the above formulation requires
modi=cation. For elastic columns, the secant stiMness at maximum response is de=ned as
the secant stiMness at the yield displacement. Following the same formulation, this results
in the column force expression given by Equation (33), where O represents the ‘ductility’
level of the elastic column which corresponds to the target displacement divided by the yield
displacement. Note that for elastic columns, this number will be less than 1. The distribution
factor is then given by the inverse of the column height for inelastic columns, and by the
fraction of the yield displacement over the column height for elastic columns.
X1 X2 O
F= (33)
L
Since it was stipulated that the column diameter and steel ratios be the same for all columns
in the bridge, the lateral strength provided will be less than the expected force given by
Equation (32) by a factor inversely proportional to the ductility level (which is ¡1 for
elastic response).
Therefore, if all section diameters and longitudinal steel ratios are the same, and the dis-
placement ductility for each column is at least one, then the column forces are inversely
proportional to the column height. This also assumes that the post-yield stiMness is zero (an
assumption of elastic–purely plastic force–deformation response). In the case where abutments

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734 M. J. KOWALSKY

provide transverse strength, the force resisted by the abutment is stipulated to be the average
of the force resisted by the columns, thus resulting in a regular force distribution. The required
abutment strength, Fab , for each abutment is then given by Equation (34), where Fsys is given
by Equation (23) and n represents the total number of columns and abutments. Therefore, the
total force resisted by the columns, Fc , can be determined by Equation (35). The total column
force, Fc can then be distributed to each of the columns in accordance with Equation (32).
If the abutment transverse strength is to be neglected, then the required total column lateral
resistance is equal to the base shear, Fsys given by Equation (23). Note that the abutment
forces given by Equation (34) are arbitrary, and that other selections are possible.

Fsys
Fab = (34)
n
Fc = Fsys − 2Fab (35)

Section design. Once the column shear forces are obtained with Equation (32), the column
moments are easily calculated. The sections are then designed to obtain the required moment
capacity at the design concrete or steel strain (whichever governs). Longitudinal reinforcement
is then provided to resist the design moment and transverse reinforcement supplied to resist
the target displacements.

Iteration. If a rigid superstructure is assumed, and the in:uence of the abutments on equivalent
viscous damping is neglected, then the procedure concludes without iteration. However, if
either of these two assumptions are not employed, then some iteration will be required.
Using the member shear forces obtained using Equation (32) and the previous target dis-
placement pattern, the secant stiMness of the columns and abutments can be obtained using
Equation (36). The revised member eMective stiMness values are then used to perform a modal
analysis to obtain a revised displaced shape. Similarly, the member shear forces can be used
to assess the equivalent viscous damping weighting factors of Equation (17).
Fi
k eMi = (36)
Oi
The entire procedure is then duplicated until convergence is reached (stability in the dis-
placement pattern from one iteration to the next). Convergence is typically reached in two or
three iterations.

SAMPLE BRIDGE DESIGNS

Overview

The design procedure shown in Figure 8, and explained in the proceeding section was ap-
plied to a series of four span bridge structures as shown in Figure 6. Each of the bridges
in Figure 6 was designed with and without abutment restraint for a total of four structures.
For the purposes of this example, only the damage control limit state de=ned by a concrete

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 735

Figure 8. MDOF direct displacement-based design :owchart.

compression strain of 0.015 and a steel tension strain of 0.06 was considered. The input was
the displacement response spectra from the Caltrans ATC-32 soil type C, magnitude 8; 0:6g
PGA spectra [15] generated for various damping levels as shown in Figure 2. For the bridges
with abutment restraint, an abutment yield displacement of 25 mm was arbitrarily selected
and a maximum damage control displacement of 75 mm was speci=ed. In practice, the abut-
ment yield displacement will be a function of the soil conditions and abutment characteristics
(wingwall vs pile supported). It is expected that after an initial design, a pushover analysis
of the abutment will reveal the force–displacement response that is utilized in the subsequent
iterations. For the purposes of this example, the abutment force–displacement response is
assumed to be bilinear.

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736 M. J. KOWALSKY

In all cases, the target displacement pattern was obtained through the rigorous modal
analysis method which considers the :exibility of the superstructure in calculating displace-
ments. From the results, it will be obvious that assuming a rigid superstructure for these
bridges would not have been an acceptable assumption.
The bridges were 200 m in length with 50 m span lengths. Column heights were con=gured
as either 7, 11, 7 m, or 14, 7, 21 m. All steel was assigned a yield stress of 455 MPa,
while concrete compressive strength was 35 MPa. The mass of the superstructure was set at
12 445 kg=m, and the strong-axis second moment of area was equal to 55 m4 . The elastic
modulus of all the concrete was 29:6 GPa. In addition to using the displacement-based design
approach, all four bridges were also designed using force-based design, where the period was
based on section stiMness equal to 50 per cent of the uncracked section stiMness. A force
reduction factor of 4 was used for the force-based design approach, consistent with current
Caltrans design recommendations.
In addition to the four comparative designs, an additional design was performed using
displacement-based design, where the span lengths of the irregular bridge were varied as 25,
25, 125, 25 m from left to right. Subsequent designs were also performed using displacement-
based and force-based designs, where the spectra employed were from real earthquakes.
Namely, the 1976 Tabas record, the 1994 Northridge Sylmar record, and the 1940 El Centro
north–south record (multiplied by a factor of 2). The displacement response spectra for these
records generated for various damping levels are shown in Figure 9.

Regular symmetric bridge with abutment restraint—Case 1

The =rst bridge considered is shown in Figure 6(a). As an example of the application of the
procedure, the design steps for this bridge are presented in detail. At the start of the procedure,
the only known information are the column heights and superstructure mass. Based on the
design experience, a column diameter of 1:5 m is assumed which allows calculation of the
yield curvature with Equation (9). Note that this selection can be revised, if required. The yield
displacement of each column is then evaluated with Equation (28). As previously established,
the abutment yield displacement has been arbitrarily selected as 25 mm. For the =rst iteration
of the procedure, the criteria for the target maximum displacement are assumed to be a drift
ratio of 3 per cent for each column and 75 mm for the abutments. In order to establish the
initial target displacement pro=le, a modal analysis is performed where the column stiMnesses
are assumed to be 10 per cent of the uncracked section stiMness, and the abutment stiMness is
equal to 30 per cent of the secant stiMness at yield. These assumptions on eMective stiMness
are meant to be initial estimates only. After the =rst iteration, the secant stiMness for the
members can be calculated directly and the modal analysis revised accordingly.
Performing a modal analysis with the assumed secant stiMness values results in a normalized
displacement pattern as shown in row 2 of Table I (marked as modal shape). Based on this
displaced shape and the strain-based target displacements in row 3 of Table I, it is evident that
the abutments are the critical elements. Therefore, the displaced shape of row 2 is then scaled
according to the abutment target displacement resulting in the target displacement pattern
shown in row 4 of Table I.
Once the target displacement pattern is established, the individual member ductility values
(Equation (1)) are calculated along with the corresponding equivalent viscous damping values
(Equation (2b)), where r = 0:05. All calculations are summarized in Table I. The equivalent

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 737

Figure 9. Real EQ Spectra used for design: (a) Tabas, (b) Sylmar, (c) El Centro N–S.

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738 M. J. KOWALSKY

Table I Design example summary—symmetric regular bridge with monolithic abutments.

Item Abut. 1 Col. 2 Col. 3 Col. 4 Abut. 5
K eM (N=m) 2.33e7 6.43e6 1.67e7 6.43e6 2.33e7
Modal shape 0.29 0.76 1.0 0.76 0.29
Initial damage-based 75 210 330 210 75
target
displacement (mm)
Target. displ. (mm) 75 197 259 197 75
Yield displ. (mm) 25 61 150 61 25
Displ. ductility 3 3.23 1.73 3.23 3
Damping (%) 16.6 17.1 11.7 17.15 16.6
System displ. 206 mm
System damp. 15.8%
System TeM 1:58 s
System MeM 2 194 000 kg (0:88Mtot )
System K eM 34 643 kN=m
System force 7153 kN
Member shear (kN) 1431 1628 1036 1628 1431
Col. moment (kN m) — 11 396 11 396 11 396 —
Steel ratio (%) — 1.56 1.56 1.56 —
N=A depth (mm) — 391 391 391 —
K eM (N=m) 1.91e7 8.26e6 3.75e6 8.26e6 1.91
Modal shape 0.41 0.79 1.0 0.79 0.41
Strain-based target 75 266 598 266 75
displ. (mm)
Target. displ. (mm) 75 147 186 147 75

Final iteration calculations

Target. displ. (mm) 75 155 208 155 75
Displ. ductility 3 2.54 1.39 2.54 3
Damping (%) 16.6 15.3 9.3 15.3 16.6
System displ. 163 mm
System damp. 13.7%
System TeM 1:17 s
System MeM 2 257 000 kg (0:91Mtot )
System K eM 64 671 kN m
System force 10 572 kN
Member shear kN 2114 2406 1531 2406 2114
Col. moment (kN m) — 16 842 16 842 16 842 —
Steel ratio (%) — 2.90 2.90 2.90 —
N=A depth (mm) — 459 459 459 —
K eM (N=m) 2.82e7 1.55e7 6.99e6 1.55e7 2.82e7
Modal shape 0.36 0.75 1.0 0.75 0.36
Strain-based target 75 232 524 232 75
displ. (mm)
Final. displ. (mm) 75 155 208 155 75 mm
Final strains 75 mm (governs) c = 0:0091 c = 0:004 c = 0:0091 75 (governs)

system displacement is then calculated with Equation (16), while the equivalent damping is
obtained with Equation (20) (for the =rst iteration—Equation (18) subsequently). The eMective
period at maximum response is then obtained with Figure 2. This is then followed by the

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 739

Table II. Summary of bridge designs: displacement-based design (DBD) and force-based design (FBD).
Case Col. Abut. DBD FBD DBD FBD Fy DBD FBD DBD FBD
height cap. col. D col. D Fu (kN) (kN) #‘ (%) #‘ (%) O O
(m) (kN) (m) (m)
1 7 2114 1.5 1.5 2406 1566 2.90 1.71 2.54 4
11 (1099 1.5 1.5 1531 598 2.90 0.75 1.39 4
7 FBD) 1.5 1.5 2406 1566 2.90 1.71 2.54 4

2 14 3244 2 2 2654 400 2.64 0.75 0.58 4

7 (1185 2 2 5308 3825 2.64 1.98 3.37 4
21 FBD) 2 2 1769 266 2.64 0.75 0.35 4

3 7 0 1.5 1.5 2283 1593 2.44 1.77 3.92 4

11 1.5 1.5 1453 423 2.44 0.75 1.39 4
7 1.5 1.5 2283 1593 2.44 1.77 3.92 4

4 14 0 1.75 1.75 1445 410 1.82 0.75 1.55 4

7 1.75 1.75 2891 1935 1.82 1.23 4.37 4
21 1.75 1.75 964 106 1.82 0.75 0.56 4

calculation of the eMective mass (Equation (21)) and secant stiMness at maximum response
(Equation (22)). The system design force is then obtained with Equation (23). The member
shear forces are then obtained with Equation (32), and the design moments calculated. Since
the column sections have equal moment demand, they will also have equal reinforcement
quantities. Damage-based displacements are then obtained through the use of Figure 5 and
Equation (7). Note that the steel strain did not govern the design of this bridge as is evident
from the trends in Figure 5. The column and abutment secant stiMness values can also be
recalculated at this point as the column forces and previous member displacements are now
known. This is then followed by a revised modal analysis with the new secant stiMness
properties resulting in a new target displacement pro=le which is again scaled according to
the critical member.
Also shown in Table I are the calculations for the =nal iteration (a total of 4 were needed
for this example). This bridge was governed by the abutment displacement criteria, therefore,
all the columns will be at a concrete strain state less than the target value of 0.015 as shown
in Table I. The required longitudinal steel ratio for the columns of this bridge is 2.90 per cent.
The same diameter columns in the force-based design bridge contain reinforcement ratios of
1.71 per cent in the exterior columns and a nominal 0.75 per cent in the central column as
shown in Table II. Also, the force-based design bridge attracts proportionally more force in
the exterior columns than the interior column since analysis is based on elastic properties
rather than secant stiMness properties.

Irregular asymmetric bridge with abutment restraint—Case 2

In the case of the irregular bridge, the results from displacement-based design indicate col-
umn reinforcement ratios of 2.64 per cent, while the force-based design bridge requires a
reinforcement ratio of 0.75 per cent (nominal) in the exterior columns and 1.98 per cent in

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740 M. J. KOWALSKY

the interior (all results are summarized in Table II). In this case, force-based design resulted
in a larger proportion of force in the stiM central column than for the bridge designed with
displacement-based design.

Regular symmetric bridge without abutment restraint—Case 3

A similar trend is noted for the bridges without abutment restraint as the stiMer column
attracts more force when using force-based design. It is also noted that the displacement-
based approach indicates the ductility level for each of the columns, while in the force-based
approach, a constant force reduction factor of 4 and the equal displacement approximation
result in an assumed ductility demand of 4 for all columns at all times. The required steel
ratio for the bridge designed with direct displacement-based design is 2.44 per cent for all
columns, while it is 1.77 per cent for the exterior columns and 0.75 per cent (nominal) for
the interior column when force-based design is used.

Irregular asymmetric bridge without abutment restraint—Case 4

For the last structure, similar results are obtained. The required steel ratio for the direct
displacement-based designed bridge is 1.82 per cent, while the force-based design bridge
requires 0.75 per cent (nominal) in the exterior columns and 1.23 per cent in the interior
column.

Irregular bridge with variable span lengths and no abutment restraint

This bridge has the same con=guration as the bridge of case 4 except that the columns have
been moved such that variable span lengths are obtained. Namely, the column separation from
left to right is 25, 25, 125, and 25 m. This results in a signi=cant dynamic mass associated with
the tall exterior column. Since all other examples have equal masses at each degree of freedom,
this example aMords the opportunity to check the procedure for a more di@cult and irregular
structure. Applying the design approach to this bridge results in a column reinforcement ratio
of 1.10 per cent, which is ¡1:82 per cent for the equal span length con=guration. Also, the
column ductility values have all increased, particularly the tall exterior column which was
previously at a ductility of 0.56, in this example, achieves a value of 1.11.

Bridges designed to real earthquake response spectra

The last series of examples were designed for real earthquake displacement response spectra
as opposed to the linear design spectra used previously. The objective of these examples is to
provide a basis for determining the accuracy of the procedure where the use of an approximate
linear spectra as shown in Figure 2 is removed as a variable and the real spectra as shown
in Figure 9 are used instead. This will become apparent during the discussion related to the
veri=cation of the procedure. For all these designs, a target concrete strain of 0.015 was
used, and since all bridges were free from abutment restraint, the strain criteria governed the
designs.
A summary of all six designs can be found in Table III. Note that the designs performed
for the El Centro record used a displacement response spectra multiplied by a factor of 2 such

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 741

Table III. Summary of design results for bridges designed with real earthquake records.
EQ Con=g. Col. D (m) O sys (mm) sys (%) TeM (s) Fsys (kN) #‘ (%)
Tabas Regular 1.75 279 16.8 1.68 8669 2.21
Sylmar Regular 1.75 309 16.1 1.47 1124 3.23
El Centro Regular 1.5 270 16.4 2.06 6003 2.44
Tabas Irregular 2 389 13.9 1.98 8459 2.04
Sylmar Irregular 2 427 13.4 1.84 10136 2.60
El Centro Irregular 1.75 323 13.3 2.16 6295 2.34

that the peak ground acceleration would be similar to the other records. For each of the two
bridge con=gurations, the largest design force was always obtained with the Sylmar input,
while the design force was smallest for the El Centro record. Also, note that the damping
was essentially constant for bridges of the same con=guration.

VERIFICATION OF DESIGN APPROACH WITH TIME-HISTORY ANALYSIS

Objective

In order to verify the accuracy of the direct displacement-based design procedure in terms of
meeting the design displacement and hence damage levels, the bridges were subjected to dy-
namic inelastic time-history analysis. Analysis was conducted with the program Ruaumoko [8]
which utilizes force–displacement hysteretic rules to characterize non-linear response. For these
studies, the Takeda degrading-stiMness model [6], which was assumed for design, was utilized
in the analysis. For each of the four structures discussed =rst, analysis was performed us-
ing 5 acceleration time histories which were arti=cially generated with the computer program
SIMQKE [16] to =t the linear design spectra of Figure 2(a). However, a certain amount of
scatter in the time histories and their associated response spectra is expected. The displace-
ment response spectra for 20 per cent damping for each of the 5 acceleration time histories
were shown in Figure 2(b). We would expect a proportional amount of scatter in the analysis
results as well. For the bridges designed to the real earthquake spectra, the corresponding
real-time histories were employed in the analysis. In all the =gures depicting the analysis
results, the target displacement pro=le from the direct displacement-based design approach
will be shown along with the results of the analysis. Analysis results represent an envelope
of maximum response. In the case of the bridges designed with force-based design, a trace
depicting the expected displacement from a spectral analysis assuming elastic properties will
be shown.

Comparative examples

Regular bridge with abutment restraint. The =rst series of analysis results is shown in Fig-
ure 10, where the regular bridge con=guration is presented. Figure 10(a) represents the results
for the bridge with the monolithic abutment to superstructure connection designed with direct
displacement-based design. Note that the target pro=le matches the analysis results reasonably
well given the scatter in the time histories used for analysis. In Figure 10(b), the time-history

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742 M. J. KOWALSKY

Figure 10. Time-history analysis results—regular bridge con=guration: (a) monolithic abutments—
displacement-based design (DBD), (b) monolithic abutments—force-based design (FBD), (c) free
abutments—DBD, (d) free abutments—FBD.

analysis of the bridge designed with force-based design is shown along with the expected
displacement from a spectral analysis assuming elastic properties. In this case, the results of
analysis using elastic properties agree very well with the inelastic time-history analysis.

Regular bridge without abutment restraint. In Figure 10(c) are shown the results for the
regular bridge without abutment restraint. In this case, the agreement is generally good with
the exception of one earthquake record which resulted in signi=cantly larger-than-expected
displacements. It is also noted that the overall shape agrees very well with the time-history
analysis. In Figure 10(d) the results for the force-based design bridge are shown. In this case,
the shape and magnitude of the displacements implied by the spectral analysis assuming elastic
properties are signi=cantly and consistently diMerent from the results of inelastic time-history
analysis, particularly at the abutments.

Irregular bridge with abutment restraint. Figure 11 represents the results of the irregular
bridge con=guration. In Figure 11(a) the results of the displacement-based designed bridge are
displayed. Note that the overall displaced shape has been captured reasonably well and that
the target pro=le lies generally through the centre of the results of the inelastic time-history
analysis. Figure 11(b) represents the results of the analysis of the force-based designed bridge.
In this case, the spectral analysis assuming elastic properties failed to capture the appropriate
displaced shape, particularly in the case of the critical short central column.

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 743

Figure 11. Time-history analysis results—irregular bridge con=guration: (a) Monolithic abut-
ments—DBD, (b) Monolithic abutments—FBD, (c) Free abutments—DBD, (d) Free abutments—FBD.

Figure 12. Time-history analysis results—irregular bridge with variable span lengths.

Irregular bridge without abutment restraint. Lastly, the irregular bridge without abutment
restrain is shown in Figure 11(c), where the target-displaced shape agrees quite well with the
displaced shape from the inelastic time-history analysis results. In the case of the force-based
designed bridge shown in Figure 11(d), the agreement is somewhat worse.

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744 M. J. KOWALSKY

Figure 13. Time-history analysis results—regular bridge without abutment restraint:

(a) Tabas—displacement-based design, (b) Tabas—force-based design, (c) Sylmar—
displacement-based design, (d) Sylmar—force-based design, (e) El Centro—displace-
ment-based design, (f) El Centro—force-based design.

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 745

Figure 14. Time-history analysis results—irregular bridge without abutment restraint:

(a) Tabas—displacement-based design, (b) Tabas—force-based design, (c) Sylmar—
displacement-based design, (d) Sylmar—force-based design, (e) El Centro—displace-
ment-based design, (f) El Centro—force-based design.

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746 M. J. KOWALSKY

Variable span length bridge

The results of the inelastic time-history analysis performed on the irregular bridge with the
variable span lengths is shown in Figure 12. Although there is a signi=cant error at the far
abutment, the overall displaced shape has been captured well and the displacement of the
critical central column is very close to the target value.

Bridges designed with real earthquake spectra

By designing and analysing a series of bridges with real earthquake spectra and associated
time histories, we can remove as a variable the linear approximation of the design spectra.
Inelastic time-history analysis was performed on the six bridges designed for real earthquake
spectra. The bridges were also designed with the force-based approach and inelastic time
history analysis performed as well.
The results of the regular bridges are shown in Figure 13. For the Tabas record, the bridge
designed with displacement-based design achieves its target displacement pro=le with reason-
able accuracy. In the case of the bridge designed with force-based design, the spectral analysis
assuming elastic properties results in an implied displacement pattern that is signi=cantly dif-
ferent from the actual displacement pattern. A similar result is noted for the bridge design to
the Sylmar record, where the expected displacements from spectral analysis assuming elastic
properties are signi=cantly lesser than the actual displacements. This is a signi=cant result
as the characteristics of the Tabas and Sylmar records are quite diMerent. The Tabas record
contains several strong motion reversals over a long period of time, while the Sylmar record
contains one damaging pulse. In the case of the El Centro record, the displacement-based
design bridge attains a displacement that is somewhat less than the target value, and for the
force-based design bridge, the agreement between spectral analysis assuming elastic properties
and the inelastic time-history analysis is excellent.
In Figure 14, the results are shown for the irregular bridge con=guration. In all three cases,
both approaches yield good agreement; however, the abutment displacements are generally
overestimated in the force-based designed bridges.

CONCLUSIONS

A procedure for direct displacement-based design of continuous multi-span bridge systems

was presented. The procedure was applied to a series of four span bridge structures, and
dynamic inelastic time-history analysis was performed to provide veri=cation. The results
indicate that the bridges designed with direct displacement-based design generally achieved
their target displacement pattern. It was also shown that spectral analysis assuming elastic
properties utilized in force-based design can at times be very accurate and at other times incur
signi=cant errors, especially when large stiMness irregularities are introduced, particularly for
systems with abutment restraint. Surprisingly, regular symmetric bridges without abutment
restraint tended to yield very inaccurate spectral analysis displacements as opposed to the
irregular con=guration.
Although the displacement-based design procedure appears promising, several areas require
further investigation. In particular, the approach used for abutments in the design process must

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 SEISMIC DESIGN OF CONTINUOUS CONCRETE BRIDGES 747

be investigated as there appeared to be the largest error at those locations. It will also be
important to conduct a parametric study in the application of equivalent SDOF systems for
MDOF bridges. Additionally, multi-frame long span bridges with expansion joints and curved
bridges have not yet been studied. In each of these cases, rigorous estimation of the inelastic
displacement patterns will be more complex than the process described here, and eMorts should
be aimed at simple methods for determining these patterns. A related area of research will
involve an investigation as to when the assumption of a rigid superstructure is valid, as such
an assumption greatly simpli=es the process for evaluating the inelastic displacement pattern.

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Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:719–747