sequential hinge formation.pdf

SR5 Lake Washington Ship Canal Bridge pushoveranalysis

p

Thomas A. Ballard*, Hassan Sedarat

SC Solutions Inc., Suite 202, 3211 Scott Blvd., Santa Clara, CA 95054, USA

Abstract

In this paper the results of SC Solutions performance-based evaluation of the as-built Unit 10 frame of the SR5Lake Washington Ship Canal Bridge are presented. This unit is a three-span frame, located between bridge piers 24

and 27 on the north approach to the bridge and extends approximately 329 ft between expansion joints. A three-dimensional model of the Unit 10 frame was developed using the computer program ADINA (ADINA UserInterface Command Reference Manual, Volume 1: ADINA Model Definition, ADINA R&D, Inc., September 1997)and ADINAs moment-curvature beam element. This paper discusses the method of evaluation of this frame, the

material and member properties used for the pushover analysis, properties of the as-built structure, and the resultsof capacity analysis of the as-built Unit 10 assembly. Several pushover analyses were performed, consisting of massproportional and mode shape lateral load profiles. One of the mass proportional lateral load analyses was selected

for presentation in this paper. This load pattern resulted in maximum shear forces in the frame columns. Followingthe pushover analysis, a nonlinear dynamic time-history ADINA analysis was conducted to estimate maximumdisplacements and to evaluate the performance of the bridge during a simulated earthquake ground motion. # 1999Elsevier Science Ltd. All rights reserved.

1. Introduction

The SR5 Lake Washington Ship Canal Bridge is

considered to be of significant importance such that its

collapse during a seismic event would be detrimental

to the safety and commerce of the people of Seattle,

Washington [2,3]. Therefore, the Washington State

Department of Transportation commissioned a retrofit

design that would result in a predicted minimum level

of performance of no-collapse during the seismic

event.

The SR5 Lake Washington Ship Canal Bridge is

located on Interstate 5 in Seattle, Washington. This is

a multi-level bridge with twelve lanes of trac, opened

in 1961 with an average trac load in excess of

150,000 vehicle per day. The bridge is 4430-foot long,

including an 1156-foot long south approach and a 980-

foot long north approach. The approach structures are

comprised of multi-span reinforced concrete frames

with box-girder deck on two levels. These approach

structures are further divided into frames or units,

which are separated from adjacent frames by expan-

sion joints located over split columns. Foundations

consist of spread footings and pile foundations.

This as-built performance-based evaluation was

commissioned in order to assess the eectiveness of

proposed seismic retrofits. These proposed retrofits

Computers and Structures 72 (1999) 6380

0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.PII: S0045-7949(99 )00022-X

pSome of the original figures for this paper were generated

with a colour-producing terminal and submitted in colour.

* Corresponding author. Fax: +1-408-486-6083.

consist of strengthening the superstructure elementsand increasing the ductility of the supporting columns,

such that all non-linearities are forced to occur atspecific locations at the top and bottom of each col-umn, thus protecting the superstructure and foun-

dation from damage. However, this capacity-protection retrofit scheme did not address the extentof damage to the columns nor did it address the rela-

tive eectiveness of each columns retrofit in protectingthe superstructure. The frame which comprises Unit 10of the north approach is a non-symmetric, non-regular

and multi-deck structure for which an evaluation usinglinear elastic solution techniques was considered unreli-able. The purpose of this evaluation is to determinethe order of formation of each plastic hinge in the

bent frames and to estimate the degree of damage eachcolumn will sustain for a predicted response of the as-built system. The results of this pushover analysis

and subsequent earthquake dynamic analysis can beextended to determine the cost associated with retrofit-ting the bridge to a specified performance level and the

resulting safety margins and to estimate the cost ofrepairs associated with a postulated seismic event.This paper discusses the method of evaluation of the

as-built Unit 10 frame, the material and member prop-erties used for the pushover analysis, properties ofthe as-built structure, and results of the capacity analy-sis of the Unit 10 assembly. The evaluation began with

an assessment of the as-built bridge, assuming that astrength upgrade for the superstructure will be requiredto satisfy the strong beam weak column strategy

adopted for this evaluation. This analysis is also usefulfor identifying the vulnerabilities of the system aftercolumn retrofits have been carried out, because jacket-

ing of the columns will not increase their moment ca-pacity significantly, but will result in a significantcurvature ductility increase and prevention of shearfailures. The sequence of plastic hinge formation was

predicted, and the resulting demand forces that will betransmitted to the superstructure were estimated.

2. Method of evaluation

SC Solutions evaluation of Unit 10 of the SR5Lake Washington Ship Canal Bridge is designed to

estimate the performance of the frame by studyingplastic curvature of the columns, the displacements ofthe decks and total base shear. The evaluation began

with construction of two-dimensional (2D) bent framemodels, which include nonlinear moment-curvaturerepresentation of all column members. The superstruc-

ture, including the box-girder deck and cross-beamswas modeled with linear elastic elements. This assumesthat these components will be retrofitted to increase

their strength, concentrating all non-linearities into thesupporting columns.

Moment-curvature properties were derived based onmaterial models discussed in the next section. Thesematerial properties were incorporated into column

cross-section models which are exercised in the BIAXprogram [7] which computes, using numerical inte-gration, the cross section moment-curvature properties,

considering concrete and reinforcing stressstrainproperties. In this manner, the columns moment-cur-vature properties are computed for specified levels of

compressive and tensile axial force, resulting in a com-plete definition of the axial force-bending momentyield surface.This yield surface is the basis for determining col-

umn hinging during the pushover analysis. The geome-try of the frames is described using linear andnonlinear finite elements for the superstructure and

columns, elastic springs for soil stiness, and rigidlinks to couple structural components.For this evaluation, it was considered important to

provide enough refinement along the length of the col-umn to capture any plasticity that may occur. Forinstance, Pier 24 columns are split at the top to pro-

vide for thermal expansion of the deck structure. Thissplit is carried to an elevation of top-of-footing plus 25ft. The column will either hinge just above the split orat the top-of-footing. In order to capture this behavior,

the entire column is modeled with moment-curvatureelements.The analysis applied dead load followed by trans-

verse force-controled pushes. This tested the modeland material behavior and demonstrated the framebehavior in a direction that could be readily under-

stood. Following this, the entire Unit 10 three-dimen-sional (3D) frame model was exercised, including allfour bent frames, with the push direction varied. Pushforces are computed based on mass distribution and

mode shapes of the system that have the predominanteective modal mass participation. For the modeshape distributions, amplitude times the lumped tribu-

tary mass of the deck and columns is the basis for therelative push force at each deck level.As the pushover analysis proceeds, the formation of

plastic hinges and the curvature or maximum strain ateach hinge is tracked for evaluation of damage.

3. Material properties

All material and section properties are based on theoriginal construction drawings [1] and recommen-dations provided by Priestley et al. [4].

Concrete and steel properties, for all members in theUnit 10 frame, were specified as 3000 psi and Grade50, respectively. Since the purpose of this evaluation is

T.A. Ballard, H. Sedarat / Computers and Structures 72 (1999) 638064

to estimate the most probable capacity of this frame,

the concrete strengths are factored by 1.5 and steel

strength is factored by 1.1. The concrete strength fac-

tor considers increase in strength for concrete aging

and conservative batching processes and is actually a

lower bound for typical bridges [4]. Fig. 1 shows the

concrete stressstrain properties used to develop col-

umn moment-curvatures properties.

Confined concrete properties are based on Manders

model [5], using the 3/8 in. steel jacket to provide con-

finement. Since the cross-section of the columns is

unique and non-standard, a computer program capable

of generating moment-curvature properties for general

sections was employed. This program, BIAX, uses a

modified KentPark concrete model for confined con-

crete. Therefore, the KentPark concrete model was fit

to the Mander model deriving the appropriate factors.

Reinforcing steel properties are computed based on

Ref. [4] recommendations. Minimum probable yield

stress is 55 ksi with ultimate strength 1.2 times yield

stress and stress at ultimate strain equal to 60 ksi. A

strain hardening plateau is considered. Fig. 2 presents

the stressstrain diagram for the reinforcing steel.

4. Section properties

Each column is evaluated for moment-curvature

based on engineering mechanics principles and material

properties discussed in the previous section. Typical

column section geometry is shown in Fig. 3 for both

as-built and retrofit configurations.

Cross-section geometry is described as an assem-

blage of rectangular sections with reinforcing placed

relative to the concrete rectangles. Each piece is subdi-

vided for the purpose of numerical integration to

determine resisting force and moment for a specified

section axial strain and bending curvature. The as-built

column properties are determined considering that all

of the concrete is unconfined. The proposed steel-

jacket retrofit will not significantly increase the column

stiness or strength at the extremities of the member,

where the plastic hinges form, and will increase the

strength and ductility only based on consideration of

the degree of confinement of the enclosed concrete.

This has been demonstrated in full-scale concrete col-

umn tests at the University of San Diego [6]. This is

due to lack of composite action between the steel

Fig. 1. Stressstrain properties.

T.A. Ballard, H. Sedarat / Computers and Structures 72 (1999) 6380 65

Fig. 2. Grade 40 reinforcing steel stressstrain properties.

Fig. 3. Typical column cross-section at footing.


jacket and concrete column at the ends. Away fromthe ends the stiness and strength will be increased due

to this composite action. Therefore, no appreciableincrease in lateral load resistance of the frame willresult from this retrofit detail. The pushover results

presented in this paper are for the as-built frame and,therefore, do not reflect the improvement in frameductility due to the retrofit detail. However, the as-

built pushover analysis can be used as a qualitativetool to evaluate the proposed capacity protection ret-rofit.

For this study, the as-built columns are thereforemodeled as unconfined concrete and the moment-cur-vature properties are developed along their entirelength. The retrofit columns are modeled as confined

concrete and the increase in stiness due to the jacketand grout is not considered.Fig. 4 is axial force-bending moment (PM ) curves,

at maximum concrete compression strains of 0.003 and0.004, which were defined as the maximum acceptablestrain, for a typical as-built and retrofit column, re-

spectively. These PM curves were the basis for deter-mining the axial forces for which moment-curvatureare computed. Five axial force values are selected for

this computation. These five axial force points include

maximum tensile axial force, maximum compressiveaxial force, and axial force associated with maximum

moment. Two additional points are selected betweenthe balanced axial force and maximum compressiveforce and balanced axial force and maximum tensile

force. This results in a relatively good representationof the PM curve for the section, as shown in Fig. 4.Fig. 5 is the resulting moment-curvature curves for

this typical column as computed by BIAX.

5. Unit 10 three-dimensional evaluation

The 3D as-built pushover analysis of Unit 10 frameis presented in this section. ADINA model details

included consideration of the deck centerline osetfrom the bent cap and cross-beam. The ends of thecolumn connections to the upper deck are oset from

the deck by one-half the deck thickness. Lower deckoset from cross-beam centerline is designed to capturetorsional moments in the cross-beam due to longitudi-

nal deck forces for the 3D model. This feature was in-corporated in the model to quantify the torsion inthe cross-beam member. Finite elements that form

the cross-beam-to-column joint and the cross-beam

Fig. 4. Typical concrete section PM curves.


Fig. 5. Typical concrete section moment-curvature.

Fig. 6. ADINA finite element model of Unit 10 frame.


member are modeled as linear elastic, since the ca-pacity protection retrofit strategy includes increasing

the ultimate strength of these components. The bentcap and deck members were also modeled as linearelastic. The finite element model is shown in Fig. 6.

This figure shows the outline of the structure as hiddenlines and the finite element elements as blue lines. Allcolumn elements were modeled using the nonlinear

moment-curvature beam element. Soil springs werealso included based on the geotechnical information.Two types of lateral load patterns for the pushover

analyses were selected. A mass proportional patternresults in higher shear forces in columns, whereasmodal pattern results in a flexural mechanism. Sincethe results are voluminous, this paper presents only

sample results from the mass proportional pushover.The evaluation began with an assessment of the

mode shapes and frequencies of the structure. Since

none of the important modes has an eective modalmass ratio of more than 75%, a lateral load patternproportional to a single mode may not capture the im-

portance of the higher modes. Therefore, a linear elas-tic response spectrum analysis was performed toobtain the load profile in terms of combination of all

modes. Lateral load pattern for this pushover analysis

was proportional to displacements of the structure de-rived based on CQC modal and SRSS spatial combi-

nations.Column element properties were derived consistent

with the methodology discussed in Section 4 for both

bending axes of the sections.

5.1. Unit 10 modal properties

Mode shapes were computed by ADINA based onthe initial stiness of the moment-curvature elements.

Figs. 710 show the first four mode shapes of the Unit10 frame as computed by ADINA. Modes 2 and 3 aretransverse modes with the south end and north end ofthe frame participating separately in their respective

modes. Mode 1 is a longitudinal mode with bothupper- and lower-deck participation. Mode 4 is alongitudinal mode with mostly lower deck partici-

pation.Since the behavior of the Unit 10 frame is complex

and not as easily described as the 2D pushovers, the

push force was determined in several ways. The firstfour pushover analyses were performed using massproportional loads. The frame and deck reactive mass

was lumped at the two deck levels and the frame was

Fig. 7. Unit 10 ADINA mode shape 11.592 s period.


pushed to collapse. The two horizontal directions werepushed simultaneously with the signs of the forces var-ied to derive four force vectors. These cases were for

positive longitudinal and positive transverse forces;positive longitudinal and negative transverse forces;negative longitudinal and negative transverse forces;

and negative longitudinal and positive transverseforces. The results of four push force profiles showedsimilar forcedisplacement relations. Therefore, detail

results are only presented for the positive longitudinaland positive transverse force case.The second method of pushing the frame was based

on an CQC modal and SRSS spatial combinations of

the first forty modes of the system with static correc-tions made for the missing higher modes. A linear elas-tic response spectra analysis was carried out using the

design spectrum.

5.2. Unit 10 mass proportional pushover results

This section presents a small fraction of the resultsof the mass proportional pushovers. The sequence of

plastic hinge formation was determined and columnshear demand forces were computed for comparison toshear capacities. Column axial forces were also com-

puted. Column bending moments were computed forinformation only, since curvature ductility is the basisfor capacity calculations. Deck axial forces, vertical

and transverse shear forces and transverse bendingmoments are also computed. Bent-cap and cross-beammoments and forces are also recovered and presented.

5.2.1. Unit 10 pushover: mass proportional

The mass proportional pushover results for the Unit10 frame are presented in this section. Fig. 11 showsthe pushover curve presented in the form of percent oftotal Unit 10 frame mass versus resultant deck displa-

cements. This curve is used to determine the total baseshear versus equivalent displacement for a targetdemand on the structure.

The collapse scenario is presented in Fig. 12. In thisfigure, each plastic hinge is shown as a red or blue cir-cle at the approximate location of hinge formation.

Red hinges represent yielding about the transverse axisof the bridge and blue hinges represent yielding aboutthe longitudinal axis of the bridge. Hinge sequencesare numbered in the order of their initial formation,

i.e., the point at which elastic curvatures are exceededin the section. The point at which a hinge exceeds itsultimate curvature capacity is indicated by placing the

sequence number for that event in parenthesis.This yielding sequence indicates that the ramp is

first to experience plastic hinging at 17% g equivalent

lateral load and collapses at about 25% g equivalentlateral load or 0.75 ft of resultant upper deck displace-ment. Pier 26 lower columns form hinges at about



Fig. 11. Mass proportional pushover curve.

Fig. 12. Pushover: plastic hinge sequence.


0.19 g equivalent lateral load. At 0.31 g equivalent lat-eral load and 1.00 ft resultant upper deck displacement,

the two east columns of Pier 26 exceed curvature ca-pacity at their lower ends. The remaining figures in thissection are samples of the resulting forces and moments

in the system as the unit is pushed. These graphs areused to determine the demands in the elastic members ofthe structure at each point in the pushover history and

to check if there is any shear failure in the columns.Figs. 1320 present typical elastic and inelastic forcesand moments.

6. Dynamic analysis of Unit 10

In a pushover analysis the response is controlled bya single pre-defined deformed shape or mode and itsshape is assumed to remain unchanged throughout the

push. Therefore, eects of higher modes may not beaddressed properly. Establishing the direction of pushin a complex 3D structure so that meaningful results

can be interpreted is not straightforward. In the fore-going pushover analyses, we used response spectrumanalysis technique as well as mass proportional load-

ings to estimate a most probable collapse mode for thepush and to include the eects of higher modes.However, when the structure undergoes nonlinearbehavior the mode shapes and frequencies changes and

the direction of the push that was obtained with thesepush load profiles will not hold in general. Due to the

complexity of the bridge, both in plan and elevation,and in order to make sure that the eects of higher

modes are taken into account properly, it was necess-ary to validate the pushover study with at least oneinelastic dynamic time-history analysis. With this type

of analysis the response of structure is not limited bythe assumed deformed shape that might be used forthe pushover analysis, because nonlinearity in the

structure will modify the response in time.

6.1. Input motions

The SR5 bridge was subjected to the S90W and ver-tical components of the 1941 El Centro groundmotions. The Peak Ground Acceleration (PGA) of the

input ground motions in longitudinal, transverse, andvertical directions were scaled to 0.29 g. Fig. 21 showsthe input ground motion time-histories.

6.2. Summary of the results

In this section, we are summarizing the resultsobtained from the time-history analysis of the bridge.Fig. 22 shows the time-history of the displacement at

the upper level of the bridge in the transverse andlongitudinal directions. The response displacementtime-histories were obtained by averaging the displace-ments of each node in the upper level deck.

The forcedisplacement relations shown in Figs. 23and 24 are total base shear versus the averaged

Fig. 13. Pushover: typical column longitudinal shear forces.


Fig. 14. Pushover: typical column transverse shear forces.

Fig. 15. Pushover: typical column longitudinal bending moments.


Fig. 16. Pushover: typical column transverse bending moments.

Fig. 17. Pushover: typical column axial forces.

Fig. 18. Pushover: typical lower cross-beam and bent-cap vertical shear.

Fig. 19. Pushover: typical lower cross-beam and bent-cap longitudinal shear.


Fig. 20. Pushover: typical lower cross-beam and torsional moments.

Fig. 21. 1941 El centro ground motion: input displacement time-histories.


Fig. 22. Response displacement time-histories at the upper level.

Fig. 23. Forcedisplacement relations in the longitudinal direction.


displacement of the upper level of the bridge. Thepushover curve obtained from the modal push is alsooverlayed on these graphs. These graphs clearly show

that the pushover curve bound the response and thatthe maximum displacement is not more than 1.5 ft.The sequence of plastic hinges were similar to thepushover analyses, confirming the results obtained

from these analyses.

7. Conclusions

Unit 10 of SR5 Lake Washington Ship Canal Bridgewas modeled with the as-built properties. Two-dimen-

sional models of bents 24 through 27 and 3D model ofthe entire Unit were developed. The results of thepushover analyses were briefly summarized in thisreport. It was shown based on Unit 10 pushover ana-

lyses that the frames ultimate displacement capacity isbetween 2.0 and 3.0 ft provided column retrofits wereimplemented to increase curvature ductility. Column

hinge sequence and locations were predicted. Axialforces, shear forces and bending moments in columnsand decks were summarized. Shear forces, torsional

moments, and bending moments of cross beam andbent cap were also summarized. These results can bereadily employed in a performance based design

method using Acceleration-Displacement-Response-Spectrum (ADRS). Performance of the force con-trolled action components and deformation con-

trolled action components can be easily evaluatedusing these results.Three-dimensional pushover analysis of structures is

a complex but useful procedure to estimate the ca-

pacity of structures. The same complexity in this analy-sis demands more careful attention in determination ofthe target displacement. Once the target displacement

is established, the extend of columns retrofits need tobe determined based on the types and locations of fail-ure.

A minimum of one 3D nonlinear dynamic time-his-tory analysis is recommended to validate the target dis-placement and retrofit design.

References

[1] Primary State Highway No. 1, Seattle Freeway, Lake

Washington Ship Canal Bridge, Washington State

Highway Commission, Department of Highways,

Olympia, Washington, 1958, pp. 1013, 1720, 2356.

[2] SR5 Lake Washington Ship Canal Bridge, Seismic

Vulnerability Assessment, Part 1, Seismic Assessment

Profile, Andersen Bjornstad Kane Jacobs, 1993.

Fig. 24. Forcedisplacement relations in the transverse direction.


[3] SR5 Lake Washington Ship Canal Bridge, Seismic

Vulnerability Assessment, Part 2, Detailed Seismic

Assessment, Andersen Bjornstad Kane Jacobs, 1993.

[4] Priestley MJN, Seible F, Calvi GM. Seismic Design and

Retrofit of Bridges. New York: Wiley, 1996.

[5] Mander JB, Priestley MJN, Park R. Theoretical stress

strain model for confined concrete. Journal of Structural

Division, ASCE 1988;114(8).

[6] Correspondence with F. Seible, April 29, 1998.

[7] Wallace JW, Moehle JP. BIAX: a computer program for

the analysis of reinforced concrete sections, Report No.

UCB/SEMM-90/12, University of California at Berkeley,

Berkeley, CA, July 1989.


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