Analysis of Steel Bridge Girders

7/30/2019 Analysis of Steel Bridge Girders

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Displacement Based Analysis of Steel Girder Bridges with

Ductile End Cross Frames

Lyle P. Carden, Ahmad M. Itani and Ian G. Buckle

ABSTRACT

Steel bridge superstructures will not necessarily remain elastic during transverse earthquake

excitation unless designed to do so. Unlike in structural concrete bridges, it is possible to use

components in the superstructure of steel girder bridges to reduce the transverse seismic forces.Recent experimental and analytical investigations showed that ductile end cross frames are

effective in reducing the transverse seismic demand. The primary objective of this paper is to

describe a displacement based analysis procedure for the design of ductile end cross frames.The displacement based capacity-demand spectrum analysis procedure used for the design of

seismically isolated bridges is adapted for the design of bridges with ductile end cross frames.

The bi-linear properties of the end cross frames are defined by the properties of the ductile bracesand transverse stiffness of the girders. In order to minimize the transverse girder stiffness and

maximize the allowable drift in the ends of the girders they are designed with no shear studs in

the regions directly above the supports, and bearings which can accommodate relatively large

rotations.Minimum and maximum bi-linear properties are used during analysis of a bridge with ductile

end cross frames to calculate the maximum drift and shear force in the cross frames respectively.

The maximum displacements may then be determined for the end cross frames using thecapacity-demand spectrum analysis procedure. Once the maximum shear force at each support is

calculated, the critical components in the transverse load path, including the shear studs, cross

frames, bearings and substructure, can be designed for the maximum force effects. Comparisons

of the response obtained from the structural analysis procedure with the response from non-lineartime history analysis showed a relatively accurate yet conservative calculation of the response.

_____________

Lyle P. Carden, Center for Civil Engineering Earthquake Research (CCEER), University of Nevada, Reno, Dept of Civiland Environmental Engineering /258, Reno, NV 89557, USA

Ahmad M. Itani, Center for Civil Engineering Earthquake Research (CCEER), University of Nevada, Reno, Dept of

Civil and Environmental Engineering /258, Reno, NV 89557, USAIan G. Buckle, Center for Civil Engineering Earthquake Research (CCEER), University of Nevada, Reno, Dept of Civil

and Environmental Engineering /258, Reno, NV 89557, USA


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INTRODUCTION

The assumption that the superstructure of a bridge will remain elastic during a large earthquake

is not necessarily valid for steel plate girder bridges, particularly for transverse excitation. Past

earthquakes, including the 1994 Northridge earthquake (Astaneh-Asl 1994) and 1995 Kobe

earthquake (Bruneau 1996), have shown considerable damage to the end cross frames, bearings,bearing stiffeners and other superstructure components. These critical components should be

designed for the effects of seismic loading. In addition certain critical components, particularly the

end cross frames (or diaphragms), can be designed to reduce the transverse seismic demand on abridge. The end cross frames are those load resisting members placed transversely between the

girders of a steel plate girder bridge, and located above the abutments or intermediate bents (or

piers). Analytical investigations and subassembly experiments have shown that ductile end crossframes are effective to varying degrees according to the ductile elements used, such as V-braces

with curved diagonals (Astaneh-Asl 1996), eccentric braced frames, a shear panel system and

triangular plate system (Zahrai 1999).In a parallel study to that reported in this paper (Carden 2004) the performance of ductile end

cross frames in a single span model of a steel plate girder bridge has been investigated using shaketable experiments. Two types of ductile end cross frames were considered, one using single angle

X-braces and the other using buckling restrained braces (BRBs) as shown in Figures 1 and 2.Both of these systems were shown to exhibit ductile performance of the bridge model and were

able to allow larger drifts, due to the geometry of the ductile members, than other systems.

Allowing reasonable drifts is the key to being able to lower the seismic forces in a bridge bylengthening its natural period. However, the drifts need to be limited to levels that provide

acceptable performance. It is recognized that the hysteresis loops for the X-braces exhibited

pinching due to strength and stiffness degradation after buckling of the single angles. Nevertheless,adequate performance was observed in the cross frames. However, the BRBs resulted in better

performance than the X-braces due to improved energy dissipation (Carden 2004). They also have

Figure 1. Bridge model with ductile end cross frames using X-braces


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Figure 2. Bridge Model with ductile end cross frames using BRBs

the advantage of full functionality after a larger magnitude earthquake compared to the X-braces.

These advantages are balanced against a cost premium. An analysis procedure is discussed in this

paper in order to implement these systems.

DUCTILE END CROSS FRAMES

Conventional seismic design assumes the substructure will yield during the design earthquake

with plastic hinges occurring in the columns. This damage will affect bridge functionality and

perhaps even require closure until repaired. Alternatively the end cross frames may be designed to

yield, thus protecting the substructure from serious damage. The advantages of this behavior arethat repair is relatively straightforward and bridge functionality is not affected.

In order for a ductile end cross frames to be effective there must be a relative transversedisplacement between the top and bottom of the girders. This mode of deformation is affected

by: 1) the transverse properties of the girders, and 2) the properties of the ductile end cross

frames. In previous studies it was assumed that the top and bottom flanges of each girder wasfixed from rotating about their longitudinal axes and that the girder deformed transversely by

bending of the bearing stiffeners (Zahrai 1999). For normal size bearing stiffeners, the stiffness

provided by these members can be considerable and can dominate the response of the ductile end

cross frames. In this study it is recognized that if the bearings are rotationally flexible and thereare no shear studs connecting the deck slab to the top flange of the girders near the end cross

frame region, then the ends of the girders are relatively free to rotate about their longitudinalaxis, resisted primarily by torsion in the girders. The end rotation provides the necessary relativedisplacement between the top and bottom flanges of the girders. This principle is used to

minimize the transverse stiffness of the girders so that the properties of the ductile end cross

frames are not dominated by the girders. In reality the bearings have some stiffness and theeffect of the shear studs varies depending on where they are terminated on top of the girders.

However, these effects may be calculated. In negative moment regions of girders it is common

in many states to omit shear studs to avoid issues with fatigue in the top flange of the girders.


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Therefore, omitting studs near the end cross frames locations is an extension of this philosophy

applied also at the abutment supports. However, some type of connection is necessary in orderto provide adequate transverse load path between the deck slab and the end cross frames. For

this purpose the top chord of the end cross frames may be connected to the deck slab through

shear studs. An example of ductile end cross frames using buckling restrained braces in a four

girder bridge, is shown in Figure 3. This example illustrates the use of a top chord connected tothe bearing stiffeners, with vertically slotted holes to allow rotation of the girders.

Figure 3. Example of BRBs as ductile end cross frames at the support of a four girder bridge.

ANALYSIS METHODOLOGY FOR A BRIDGE WITH DUCTILE END CROSS FRAMES

The approach typically used for the seismic analysis of bridges in the AASHTO (1998) and

CALTRANS Seismic Design Criteria (2001) is based on the elastic analysis of a bridge.

AASHTO (1998) then modifies the forces using a response modification factor in order to allowsome ductility in elements of typically the substructure. CALTRANS (2001) uses the elastic

displacements and checks the ductile capacity of the substructure elements. These

methodologies are primarily based on the equal displacement rule in which the inelasticdisplacement demand in a structure is assumed to be the same as the elastic displacement

demand. This rule is typically valid for longer period structures but is not valid for structures

with shorter periods. This recognized in many publications such as the ATC 32-1 (ATC 1996).

Bridges with relatively flexible column bents usually have properties in the range for which the


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equal displacement rule can apply, however bridges which are supported on relatively stiff piers

and abutments may not necessarily respond in this manner. Flexibility in the superstructure andsubstructure provide another complication in calculating the displacement demand in the end

cross frames using an elastic analysis (Alfawakhiri 2001).

In recognition of these and other limitations of the traditional methodology, the capacity-

demand spectrum analysis approach, commonly used in the analysis of seismically isolatedstructures (AASHTO 1999), is proposed for the analysis of steel plate girder bridges with ductile

end cross frames. It is recognized that bridges with ductile end cross frames have shorter natural

periods than isolated structures, but, the capacity-demand spectrum procedure is still applicable.The procedure is able to capture the effect of: 1) additional damping in the structure provided by

the ductile end cross frames; 2) post-yield stiffness of components in the load path of the ductile

end cross frames, and 3) allow for flexibility of the superstructure and substructure. Mostimportantly the procedure is displacement based and results in a direct evaluation of the

displacement demand in the end cross frames, which may then be compared to the displacement

capacity. The displacement capacity of the ductile end cross frames is determined by the capacityof the girders and the capacity of the braces in the cross frames.

When applied to a seismically isolated bridge, the capacity-demand analysis procedure modelsthe non-linear isolators as bi-linear. The same approach is proposed for modeling of the non-linear

ductile end cross frames when these are used in a bridge. The bi-linear model reflects the transversestiffness of the girders and the non-linear properties of the ductile mechanism.

DEFINITION OF BI-LINEAR PROPERTIES

Properties of Girders

The transverse stiffness of a girder at a support can be modeled as shown in Figure 4.

Figure 4. Model of the end of the girder


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In this figure the member represents the bearing stiffener to which the end cross frames are

attached and is modeled by a column with a flexural stiffness defined by EIws of the bearingstiffener and a girder depth, dgf, between the centroids of the top and bottom flanges. At the top

of the girder a rotational spring represents the torsional stiffness of the top flange (ks). At the

base of the girder another rotational spring represents the rotational stiffness of the bearing (kb).

A shear spring is used to model the torsional stiffness of the girder (kt).The torsional stiffness of the girder depends on the length of the girder, L1, between the end

cross frame location and nearest intermediate cross frame (Fig. 5) and can be calculated as

(Brockenbrough 1968, Carden 2004):

=

a

L

a

Lad

GJk

gf

g

t

11

2

tanh

1(1a)

or at the support of a continuous section of a girder, where L1 is the distance between the support

and intermediate cross frames in each direction from the support cross frames (Fig. 5), by:

=

a

L

a

Lad

GJk

gf

g

t

2tanh

2

1

11

2(1b)

where: Jg is the torsional moment of inertia for the girder, G is the shear modulus, and a is given by:

g

gygf

GJ

EIda

2= (2)

where: Igy is the moment of inertia of the girder about its vertical axis, E is the elastic modulus and

G is the shear modulus of steel.

As there are no studs immediately above the bearing stiffener, the rotational spring at the topof the girder is dependent on the torsional stiffness of the top flange over the length in which

there are no shear studs, s1 (and s2 if at a continuous support)(Fig. 5), therefore ks is given by:

+=

21

11

ssGJk fs (3)

where: Jf is the torsional moment of inertia of the top flange. This assumes there are nointermediate stiffeners between the end cross frames and first intermediate cross frame.

The stiffness of the spring at the base of the girder is determined by the rotational stiffness of

the bearing. Fully vulcanized reinforced elastomeric bearings were found to be effective at the

base of the girders as they were able to allow the necessary rotational capacity. The rotationalstiffness of these bearings can be calculated from the bearing properties.


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Figure 5. a) end of simply supported girder, b) end of continuous girder

The equivalent transverse stiffness of each girder at each support, Kg, can be calculated by

incorporating all the components which contribute to the girder stiffness, including the torsionalgirder stiffness, shear studs, bearings and web stiffeners, as shown in Figure 4, thus Kg is given

by:

)1(12

3+=

gf

wstg

d

EIkK (4)

where: is given by:

gf

ws

gfws

ws

s

gfws

b

gf

sb

gf

ws

d

EIdEI

EI

k

dEI

k

d

kkd

EI

444

)(312

+

+

++

=

(5)

If the springs, ks and kb have a stiffness equal to zero then is equal to 1 and the equivalentgirder stiffness in Equation 4 is equal to the torsional stiffness of the girder, kt. If kb approaches

a large value and ks is equal to zero, or vice versa, then is equal to3/4 and the equivalent girder

stiffness is equal to kt plus 3EIws/dgf3. This is equivalent to web stiffeners pinned at one end and

fixed at the other end. If both ks and kb approach a large value then is equal to zero, for webstiffeners fixed at both ends (12EIws/dgf

3).


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The bending moments in the flanges due to torsion of the girders may need to be checked at

intermediate supports of continuous girders. The maximum bending moment in the bottom flangeof a continuous girder Mefb is given by (Brockenbrough 1968, Carden 2004):

ttefb d

a

LakM ))

2

(tanh(83.0 1= (6)

where: di is the displacement of the top flange at the ductile cross frame location. This can becombined with the effects of gravity loading on the girder in the same way that wind loading is

combined with gravity loading for corresponding load combinations according to AASHTO (1998).

As an example, Equation 6 was applied to the bridge shown in Figure 3, with L 1 of 6.0 m and astress level in the girder of 36% of its ultimate stress level during gravity loading associated with the

earthquake load case. The calculated maximum drift in the girders was calculated to be 5.3%

(Carden 2004), where the drift is defined by the displacement of the top chord divided by the heightof the girder between the centroids of the flanges. If this drift were to be exceeded in the girders

then a plastic hinge would start to develop in the flange of the girder. Although this should not

result in collapse, in order to minimize damage to the girder, it is recommended that this be used asa design limit. Effectively pinned supports at the ends of girders will typically be able toaccommodate large drifts without significant bending moments in the flanges of the girders, and

therefore the allowable displacement at the ends of girders is unlikely to be governed by this limit

state.The bridge model was subjected to a maximum measured drift of 7.0% (Carden 2004), at the

intermediate support in a continuous two span configuration, without any observed damage to the

girders. As this is the limit of experiments, P-delta and other effects have not been consideredbeyond this level of drift. Therefore 7% is recommended as the maximum design drift.

Properties of the Ductile Braces

The transverse stiffness of the ductile elements of the end cross frames is defined by the

geometry of the end cross frames and the bi-linear properties of their components. This is

described using the properties of a BRB as an example. The axial stiffness of the BRB using thedeformable length of the brace, which is given by:

sc

sc

ubL

EAk = (7)

where: Asc is the area of the steel core, and Lsc is the effective length of the steel core. The

transverse stiffness of the ductile end cross frames is defined as the transverse shear force

divided by the transverse displacement of the top flange of the girder relative to the bottomflange. The transverse stiffness is related to the axial stiffness using the transformation, t, to

convert the transverse deformation of the end cross frames to the axial brace deformation. This

transformation is not just dependent on the slope of the brace but also the ratio of h1 to dgfas shownin Figure 3. The transverse stiffness is equal to the axial stiffness multiplied by the square of the

transformation factor. For a configuration, as shown in Figure 3, the axial displacement is

calculated to be 52% of the girder transverse displacement. The transverse brace stiffness istherefore 27% of the axial stiffness.


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The post yield stiffness, kdb, in bi-linear modeling of the BRB can be modeled as 5% of the

initial stiffness as shown from experiments with braces in the bridge model (Carden 2004). Theaxial yield strength, Pyb, for the braces is given by the expected yield force for the steel in the

core multiplied by the area of the core. The characteristic strength, Qdb, defined as the intercept

of the bi-linear hysteresis loop with the y-axis, is calculated from the other properties by:

)1(ub

db

ybdbk

kPQ = (8)

The axial yield forces and characteristic strength can be converted to transverse componentsby multiplying by the transformation factor, t. The total characteristic strength in a set of ductile

end cross frames is given by the summation of strengths based on the number of braces, nb, in the

system, giving the transverse characteristic strength of:

dbbd QtnQ = (9)

The design stiffness for a set of ductile end cross frames, kd, which is the post-yield stiffness

of the bi-linear system is given by the summation of the transverse post-yield stiffness for each

of the braces and the summation of the girder stiffnesses, therefore:

dbbggd kntKnK2+= (10)

The displacement capacity of the BRBs can be calculated from the design axial strain in the

braces. From experiments on the unbonded braces an axial strain in the braces of 2.5% isconsidered appropriate for their design as this is one third less than the ultimate strain recorded in

the braces during increasing amplitude cyclic experiments (Carden 2004). For the configuration

shown in Figure 3, this results in a 6.7% drift in the girders,

Variation of Bi-linear Properties

The AASHTO isolation guidelines (AASHTO 1999) require that an isolated structure beanalyzed for two sets of isolation properties to allow for the possible range in these properties due

such factors as material variation, manufacturing tolerances, aging, and others. A similar

philosophy is proposed for the design of ductile end cross frames. The set of minimum bi-linearproperties are used to calculate the maximum displacement response of the cross frames ensuring

that the displacement is within the capacity of the braces and the girders. The maximum bi-linear

properties are used to calculate the maximum forces in the end cross frames for which the

connections and other critical elements can be designed. Factors proposed to vary the expected bi-linear properties in order to calculate the maximum and minimum levels are given in Table 1.

These factors are based on the variation between expected properties and measured properties from

the experiments and finite element analysis. They allow for such factors as strain rate effects,friction associated with the rotation of the connections in the top and bottom chords, compression

over-strength in the BRBs, variation in the girder stiffness and fixity of the BRBs.


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Table 1. Variation factors for bi-linear properties of ductile end cross frames using buckling

restrained braces.Variable Property Minimum Maximum

Qd Material (without coupon test)1 0.9 1.1

Material (with coupon test) 1.0 1.0

Strain Rate 1.0 1.15

Friction in Chord Members 1.0 1.1

Compression Overstrength 1.0 1.1

Kd Girders2 0.8 1.2

UB Pinned2 1.0 1.5

UB Fixed2 1.0 2.5

Notes: 1. Based on properties for Nippon Steel LYP 225.

2. These factors are to be applied to the corresponding components of the post-yield stiffness

independently.

ANALYSIS AND DESIGN PROCEDURE

Design of the Bridge for Non-Seismic Loads

A flowchart outlining the analysis and design procedure is shown in Figure 6. The first step

in the procedure is to design the superstructure including the girders, deck slab, shear studs,

stiffeners and intermediate cross frames for non-seismic loads as these components do notrequire detailed seismic design. The superstructure stiffness can be determined from the

transformed superstructure properties.

Having designed the major superstructure components for non-seismic loads it is necessary

to define the seismic hazard for a bridge in order to determine an appropriate methodology fordesign of the remaining superstructure and substructure components. In higher seismic zones,

such as AASHTO zones 3 and 4, the lateral seismic forces are likely to govern the design of the

substructure and some critical components in the superstructure. In order to define the seismicinput according to AASHTO (1999) the acceleration coefficient, A, and site coefficient, S, are

determined to define the design spectrum. The importance category, I, is also used to define the

combination of variability factors required for design.

Structural Analysis

Ductile end cross frames may be selected as the ductile components in the transverse response

of the bridge for reasons previously discussed. Using the calculated properties for thesuperstructure, substructure and end cross frames, a bridge is analyzed to determine whether or not

the cross frames are adequate. The procedure is an iterative procedure whereby the cross frame

displacements are estimated, then analysis performed to check these displacements, upon which thedisplacement estimate can be revised and accuracy of the analysis improved until the estimated and

calculated displacements converge. The steps in the analysis procedure are:


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Design Superstructure (Girders, Deck, Shear Connectors, Stiffeners, Intermediate

Cross Frames) for Non Seismic Loads

Select Ductile Components from Columns, Pier Walls, End Cross Frames or Isolators


Estimate Transverse Superstructure, Column and Abutment Properties - Define

Maximum and Minimum Bi-linear Properties for End Cross Frames

Are End Cross

Frames Adequate?

Determine:

Acceleration Coefficient, A; Bridge Importance Category, I;

Seismic Performance Zone; Site Coefficient, S; (AASHTO Isolation Spec.)

No

Estimate Design Displacement in Cross Frames at each Support

Calculate Maximum Force, Fm; Effective Stiffness, K

eff; Effective Damping, ; in Cross Frames

Are Displacements

Equal to Estimated

Displacements?

Yes

Maximum PropertiesMinimum Properties

No

Determine Fmax at Column Bents and Abutments from Analysis with Maximum

Properties (AASHTO 3.10.7.1, 3.10.9.4)

Yes

Are Displacements

Equal to Estimated

Displacements?

No

Design Top Chord and Shear Studs for

Fmax

Design Cross Frame Connections for Fmax

Design Substructure for Fmax

Design Bearings and their Connections forFmax

Use Model to Evaluate:

Effective Period, Teff; Effective Damping, ; Damping Factor, B; Seismic Coefficient, Csm for EntireBridge. Determine Displacement in Cross Frames at each Support

Design Superstructure (Girders, Deck, Shear Connectors, Stiffeners, Intermediate

Cross Frames) for Non Seismic Loads

Select Ductile Components from Columns, Pier Walls, End Cross Frames or Isolators


Estimate Transverse Superstructure, Column and Abutment Properties - Define

Maximum and Minimum Bi-linear Properties for End Cross Frames

Are End Cross

Frames Adequate?

Determine:

Acceleration Coefficient, A; Bridge Importance Category, I;

Seismic Performance Zone; Site Coefficient, S; (AASHTO Isolation Spec.)

No

Estimate Design Displacement in Cross Frames at each Support

Calculate Maximum Force, Fm; Effective Stiffness, K

eff; Effective Damping, ; in Cross Frames

Are Displacements

Equal to Estimated

Displacements?

Yes

Maximum PropertiesMinimum Properties

No

Determine Fmax at Column Bents and Abutments from Analysis with Maximum

Properties (AASHTO 3.10.7.1, 3.10.9.4)

Yes

Are Displacements

Equal to Estimated

Displacements?

No

Design Top Chord and Shear Studs for

Fmax

Design Cross Frame Connections for Fmax

Design Substructure for Fmax

Design Bearings and their Connections forFmax

Use Model to Evaluate:

Effective Period, Teff; Effective Damping, ; Damping Factor, B; Seismic Coefficient, Csm for EntireBridge. Determine Displacement in Cross Frames at each Support

Figure 6. Analysis and design procedure for ductile end cross frames in a steel girder bridge using acapacity-demand spectrum analysis


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1. Estimate displacements in end cross frames,2. Calculate transverse shear force in end cross frames,3. Estimate effective linear properties (stiffness and damping) in end cross frames,4. Use model to perform seismic analysis of structure with the effective properties.5. Check displacements in end cross frames and perform further iterations if necessary.

The minimum bi-linear properties are first used to calculate the maximum displacement

response. The cross frames at each support will typically have a different response, however for thefirst iteration, the maximum allowable drift may be assumed in the cross frames at each support.

The maximum drift is calculated from the displacement capacity of the girders or the displacement

capacity of the ductile braces, as previously discussed.

Having estimated the design displacement, the force in the ductile end cross frames can becalculated using the bi-linear properties. The effective stiffness of the end cross frames can then be

calculated and the effective damping in the end cross frames as per the AASHTO isolation

guidelines. A structural model of the bridge is then used to calculate the transverse seismicresponse, including the effects of superstructure and substructure flexibilities and displacement

response at each of the end cross frames. The procedure uses using the likes of an elastic multi-mode response spectrum analysis.The response spectrum used to calculate the seismic demand on the model is based on the

design spectrum and modified to reflect the calculated level of damping in the structure based on the

B-factor (AASHTO 1999). To account for substructure flexibility the AASHTO isolationguidelines adjust the equivalent viscous damping calculated in the isolation system to obtain the

equivalent viscous damping in the structure, using the relative displacement in the isolation system

compared to the overall displacement in the structure. The isolation guidelines assume no damping

in the substructure and do not consider superstructure flexibility. In this paper the isolationguideline methodology is generalized to include superstructure flexibility as well as substructureflexibility and allow some nominal damping in the superstructure and substructure. If each span of

the bridge is considered individually, for a given span, the effective damping in the span, s, can beapproximated by:

t

subsubererii

sd

ddd

++=

supsup(11)

where: i is the average calculated equivalent viscous damping in the end cross frames at each endof the span; di is the displacement in the end cross frames, also averaged between the two ends;

dsuper is the relative superstructure displacement, calculated by the maximum superstructure

displacement relative to the end displacements at the top of the girders; dsub is the averagesubstructure displacement at the end of the span; dt is the maximum total deformation in the span

including superstructure, substructure and end cross frame deformations, and; superand sub are anyeffective damping associated with the superstructure and substructure respectively. The resulting

effective damping in the structure, t, can be approximated by averaging the effective damping ineach span. This will be shown to have a significant effect if there is a relatively flexible

superstructure or substructure compared to the ductile end cross frames, which will limit the

effectiveness of these members.

The calculated displacements in the ductile cross frames from the structural analysis arecompared to the displacements estimated at the beginning of the analysis. If the resulting


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displacements are larger than the estimated displacements, then the braces need to be modified. If

more accuracy is required further iterations can be performed. Once the maximum displacementrequirements have been satisfied, a similar analysis can be used to calculate the maximum forces in

each of the ductile end cross frames using the maximum bi-linear properties for the cross frames.

Comparisons of the response from the structural analysis procedure with that from spectrum

compatible non-linear time history analysis showed a relatively accurate yet conservativecalculation of the response, with error attributed to the B-factor used for calculating damping.

Design of Critical Components for the Maximum Forces

The critical components which should be designed to withstand the maximum forces in the

bridge with ductile end cross frames are:

Connection between the deck and the end cross frames (top chords and studs), Ductile brace connections, Bottom chords,

Bearings and their connections, Abutments, piers and foundations.

With no shear studs on top of the girders at the ductile cross frame locations, a connectionbetween the deck and the cross frames is necessary using top chords. Studs on the top chords are

designed for the maximum shear in the end cross frames. The connections between the top chords

and the bearing stiffeners are also designed for the maximum shear at each end of each member.The ductile brace connections are designed using capacity design principles to be stronger than

the maximum force in the brace.

Investigating the load path in the end cross frames suggest that the bottom chords would not benecessary if the transverse loads in the bearings do not cause any damage to the bearings or

transverse restraints. However, the forces in the bearings are not uniform and the potential fordamage to a bearing restraint is relatively high. Therefore using bottom chords allows redistributionof the forces to the other bearings, if necessary. It helps to ensure that all bearings act together in

resisting transverse shear forces. The bottom chords and their connections can be designed for

forces assuming that a bearing fails and requires the load to be redistributed to the other bearings.

The bridge model exhibited shear forces in one bearing of up to two times the forces in theadjacent bearing (Carden 2004). While it is difficult to extrapolate these results to a bridge with

more girders, it is apparent that the bearings should be designed for transverse forces in excess of an

equal distribution of shear forces in the bearings. The bearings should also be designed with at leastlimited ductile action so that if overloaded the bearings can redistribute loads without failure. The

bearings should also be designed for the axial loads resulting from overturning of the bridge

superstructure due to an eccentricity between the center of mass and top of the bearings where theloads are resisted. The bearings are also subjected to rotations from the deformation of the ductile

end cross frames, therefore a combination of effects should be considered in designing the bearings.

The abutments, piers and foundations are also designed for maximum shear calculated in theductile end cross frames. The isolation guidelines (AASHTO 1999) suggest that limited R-factors

can be used to reduce the forces in ductile substructure components, as essentially elastic behavior is

still probable with the use of an isolation system. A similar approach could be adopted when using


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ductile end cross frames, although as suggested in the isolation guidelines, caution should be

exercised when applying these R-factors.

SUMMARY AND CONCLUSIONS

Ductile end cross frames can be an effective way of reducing the transverse seismic demand

in a steel plate girder bridge. While a number of options are available when using ductile endcross frames, BRBs are particularly effective as they have a relatively large displacement

capacity.

The displacement based capacity-demand spectrum analysis procedure used for the design of

seismically isolated bridges is effective when adapted for the design of ductile end cross frames.The bi-linear properties of the end cross frames are defined by the properties of the ductile braces

and transverse stiffness of the girders. In order to minimize the transverse girder stiffness and

maximize the allowable drift in the ends of the girders they are designed with no shear studs inthe regions directly above the supports, and bearings which can accommodate relatively large

rotations.Minimum and maximum bi-linear properties are used for the ductile end cross frames tocalculate the maximum drift and shear force in the cross frames respectively. Once the

maximum shear force at each support is calculated the critical components in the transverse load

path, including the shear studs, cross frames, bearings and substructure, can be designed for themaximum force effects. Comparisons of the analysis procedure with non-linear time history

analysis showed a relatively accurate and conservative calculation of the response.

ACKNOWLEDGEMENTS

The authors would like to thank CALTRANS and FHWA through the highway project at

MCEER for funding of this project. They would also like to acknowledge Dynamic Isolation

Systems, Nippon Steel and Seismic Isolation Engineering for their valuable contributions.

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Analysis of Steel Bridge Girders

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