Direct Displacement –Bad Seismic Design
of Reinforced Concrete Arch Bridges
Easa Khan 1;Timothy J.Sullivan 2;and Mervyn J.Kowalsky 3
2013年6月四级Abstract:This paper extends the direct displacement –bad design (DDBD)procedure,which was developed for buildings and conventional bridges,to the special ca of RC deck arch bridges.New design expressions are formulated for the yield drift and deformation capacity of bridge piers ated on arches.The propod methodology is applied to three ca study deck arch bridges in both the longitudinal and transver directions,and the designs are validated by nonlinear time-history analys.The results indicate that the propod methodology is capable of capturing the deck displacement and pier chord rotation within a reasonable degree of accuracy,although the respon of the arch bridge is complex and can be affected by higher modes.The rearch reveals that the arch displacement may be underestimated by the DDBD procedure,but becau the arch displacements are very small in comparison with the deck displacement,the DDBD procedure is still successful in controlling peak chord rotation demands on the bridge piers.DOI:10.1061/(ASCE)BE.1943-5592.0000493.©2014American Society of Civil Engineers.
Author keywords:Direct displacement –bad design;Seismic design of deck arch bridge;Nonlinear time-history analys.
Introduction
Becau an arch resists gravity loads in compression,a RC arch bridge can be en as an elegant and effective means of bene fitting from the high compression strength of concrete.The concrete arch bridge has a history of more than 200years (Chen and Ye 2008;Radic et al.2008);however,there were few RC arch bridges constructed until the end of nineteenth century.In the twentieth century,with the development of high-strength materials and erection technologies,large-span RC arch bridges were constructed.Concrete arch bridges can be classi fied into three main forms:deck arch,light deck arch,and half-through arch (Chen 2008).Among the types,the deck arch bridge (Fig.1)is the most commonly ud and is the main focus of this study.
The arch of deck arch bridges can be realized using different cross-ctional con figurations;single or multicell box ctions,box ribs,solid ctions,and concrete-filled steel tubes (Chen and Ye 2008).Box cross ctions and box ribs are much more popular for u in concrete arch bridges becau of their excellent rigidity and capacity of resisting bending,especially for long span bridges.T
he piers of deck arch bridges are realized either with pairs of compact rectangular RC columns aligned along each edge of the deck or rectangular wall-type piers arranged with their long axis in the transver direction of the deck.This paper focus principally on deck arch bridges with pairs of rectangular piers,but the situations should also be applicable to bridges with wall-type piers.Regarding
the arch axis,most existing bridges have adopted catenary curves as their axis,whereas others u parabolic curves.Statistics (Chen and Ye 2008)demonstrate that most bridges have a ri-to-span ratio between 1:5and 1:8,with 1:6being the most common ratio.To reduce the height of the spandrel columns (columns resting on the arch),a smaller ri-to-span ratio is an advantageous choice in deck arch bridges (Chen and Ye 2008).The Wanxian Yangtze River Bridge,with a clear span of 420m,is a good example of a deck arch bridge.
Seismic design guidelines with speci fic indications for deck arch bridges appear to be relatively limited.There are,however,veral publications in the literature that report on the ismic design and analysis of real deck arch bridges (Kawashima and Mizoguchi 2000; Zderi c et al.2007;Savor et al.2008;Franetovi c et al.2011).Current ismic design practice for such bridges appears to u the modal respon spectrum method (with pushover analys sometimes being ud to asss perfor
mance).The writers did not find clear indications in their review of the literature as to what the preferred inelastic mechanism should be for RC deck arch bridges.Given the important role of the arch and deck to the gravity-load –resisting system,and the dif ficulty in repairing such elements,it is recom-mended that the elements be designed to remain elastic,using a suitable capacity design philosophy.In contrast,a ductile plastic mechanism for a deck arch bridge could be provided by flexural yielding of the RC piers;this tends to be the preferred mechanism for more traditional RC bridges (Priestley et al.1996;Kowalsky 2002).In the bridge con figuration in Fig.1,it is clear that pier heights vary considerably along the length of the bridge.The short piers atop the arch should be expected to posss only limited deformation capacity and could be critical to the bridge performance (Franetovi c et al.2011).To ensure adequate performance of the piers,three possibilities could be considered:(1)relea the piers from the deck in the longitudinal direction so they are not subject to large de-formation demands (Savor et al.2008);(2)inrt ismic isolation devices between the piers and the deck to transfer a limited amount of lateral load to the piers;or (3)detail the piers to respond in elastically in flexure by specifying a minimum pier aspect ratio of 3.0,together with adequate transver reinforcement,and then provide the bridge
1bec报名官网
Ph.D.Student,North Carolina State Univ.,Raleigh,NC 27695-7908(corresponding author).E-mail:ekhan2@ncsu.edu 2
pingyangAssistant Professor,Dept.of Civil Engineering and Architecture,Univ.of Pavia,27100Pavia,Italy.E-mail:timothy.sullivan@unipv.it 3
Professor of Structural Engineering,North Carolina State Univ.,Raleigh,NC 27695-7908.E-mail:kowalsky@ncsu.edu
Note.This manuscript was submitted on July 16,2012;approved on March 22,2013;published online on April 1,2013.Discussion period open until June 1,2014;parate discussions must be submitted for individ-ual papers.This paper is part of the Journal of Bridge Engineering ,Vol.19,No.1,January 1,2014.©ASCE,ISSN 1084-0702/2014/1-44–58/$25.00.
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suf ficient lateral strength and stiffness to limit the deformation demands on the piers to acceptable values.The bene fit of the first solution is that damage to the short piers can be avoided,but the tradeoff is that the deck has much less restraint in the longitudi-nal direction.Conquently,large longitudinal movements must be accommodated at the abutments.To limit longitudinal movements for the Krka River Arch Bridge,Savor et al.(2008)reported that dampers were installed between the deck and abutments.The cond solution is also viable,but the cost and experti required for the u of isolation devices can be discouraging.The third solution requires only standard construction technology and takes advantage of the naturally high stiffness offered by the arch.In addition,the energy dissipated by the yielding piers helps mitigate the longi-tudinal displacements of the deck,as evident in the following summary of design results.Although concrete spalling and inelastic demands on the piers are likely to mean that repairs would be required after an inten earthquake event,the possibility of such loss may be deemed acceptable to the bridge owner if it means lower initial construction costs.With the preceding comments in mind,this paper investigates the extension of the direct displace-ment –bad ismic design (DDBD)procedure by Priestley et al.(2007)to deck arch bridges in which piers are detailed to allow inelastic respon.
Fundamentals of Direct Displacement–Bad Design The DDBD procedure,developed principally by Priestley et al.(2007),aims to overcome de ficiencies and limitations with tradi-tional force-bad code methods.This procedure provides a means of design for a structure to reach a predetermined displacement when subject to an earthquake that is consistent with design-level ismic intensity.In contrast to a force-bad design approach,DDBD attempts to design a structure that would achieve,rather than be bounded by,a given performance limit state under a given ismic intensity.This potentially permits the realization of uniform-risk structures,which is philosophically compatible with the uniform-risk ismic spectrum incorporated into most design codes.The DDBD procedure is well developed for many structural types,such as RC,masonry,and timber buildings,as well as traditional RC bridges.Interested readers should refer to the text by Priestley et al.(2007)or the model code for DDBD (Sullivan et al.2012)for more details.This ction of the paper provides a brief overview of the methodology,and the next ction details a means of extending it to deck arch bridges.
The fundamental concepts of the DDBD procedure are illustrated in Fig.2,which makes reference to a frame building,but the same approach applies for bridges.DDBD is a respon spectrum –bad approach.It differs from traditional respon spectrum methods becau it us the substitute structu
re concept developed by Gulkan and Sozen (1974)and Shibata and Sozen (1976).This concept
reprents the respon of a multidegree-of-freedom (MDOF)non-linear system with an equivalent linear single-degree-of-freedom (SDOF)system [Fig.2(a)],which are characterized by an equivalent elastic cant stiffness to the maximum respon point [Fig.2(b)]and a consistent equivalent viscous damping value that is dependent on the expected ductility demand [Fig.2(c)].In addition,ismic hazard is reprented using the displacement respon spectrum [Fig.2(d)],rather than the acceleration respon spectrum,becau the objective of the design procedure is to limit the displacements.
The DDBD procedure starts by identifying the design displace-ment,D d ,which will satisfy one or more criteria for the performance level under consideration,with possible consideration of material strain limits,displacement ductility demands,or code-speci fic drift ratios.By computing the ductility demand at the design displace-ment limit,an equivalent viscous damping value,j eq ,is obtained [Fig.2(c)]and ud to scale the elastic design displacement spectrum to the design damping value.Priestley et al.(2007)recommend the u of the following damping-dependent scaling factor,which can be found in Eurocode 8[European Committee for Standardization (CEN)1998]:
h ¼
0:070:02þj eq
!0:5
(1)
The displacement respon spectrum is entered with the design displacement to the interction with the appropriately damped re-spon curve and to find the required effective period,T e ,as shown in Fig.2(d).This is then ud together with the effective mass,m e ,of the SDOF system to obtain the required effective stiffness,K e ,as shown in Eq.(2)
K e ¼
2psonia
T e
2m e (2)urgent怎么读
The design ba shear is then found by multiplying the effective stiffness by the design displacement,as shown in Eq.(3)
英语四级多少分能过V b ¼K e D d
(3)
If P -D effects are signi ficant,they can also be accounted for through an additional ba shear component,but this term is omitted here for simplicity.The design ba shear is then distributed to the structure as a t of equivalent lateral forces,F i ,given by Eq.(4),and analysis is undertaken to obtain the required strengths of the plastic hinge regions
F i ¼m i D
i P n i ¼1i D i
V b
(4)
Fig.1.Elevation and ction view of a typical RC deck arch bridge:(a)elevation view;(b)ction A-A
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where n 5total number of lumped mass locations;and m i and D i 5mass and displacement at point i of the MDOF structure,respectively.Indications as to how to t the displacement pro file are provide
d in the step-by-step design procedure described in the next ction.
Capacity design procedures should then be implemented to ob-tain design forces for all capacity design protected members and actions.
The procedure is therefore relatively simple,with the main challenge being identi fication of the equivalent SDOF system properties.One of the main objectives of this paper is to show whether the DDBD procedure propod by Priestley et al.(2007)could be extended to arch bridge structures,as it could be expected that the complex higher mode respon of arch bridges may require something other than a simpli fied single mode procedure.In the following ctions,an approach for DDBD of deck arch bridges is propod for the transver and longitudinal excitation directions.In the last part of the paper,the promising performance of the new methodology is illustrated through examination of three different ca study bridges.
Direct Displacement –Bad Design of Deck Arch Bridges
The propod DDBD procedure for RC deck arch bridges is described parately for the transver and longitudinal respon directions in the following subctions.The design in the transver respon di-rection is arguably more challenging becau of uncertainty in the
displaced shape of the bridge,but the same general procedure illus-trated in the flowchart of Fig.3is applicable in both excitation directions.The procedure is an extension of the methodology for traditional MDOF bridges developed by Calvi and Kingsley (1995),Kowalsky (2002),and Priestley et al.(2007),with speci fic mod-i fications required to account for the effects of the arch.Direct Displacement–Bad Design of Deck Arch Bridges in the Transver Direction
The DDBD procedure of Fig.3is explained step-by-step for the transver respon direction.
Step 1:Choo Inelastic Displacement Pro file and Internal Force Distribution
Given that the displacement pro file is required to convert the MDOF bridge into an equivalent SDOF system,the first step is to choo the inelastic displacement pattern.This involves hypothesizing the lo-cation of the critical element,which is likely to be the shortest piers becau the inelastic action is assumed to be con fined to the columns of the bridge.For the transver respon direction,both the deck and arch could be expected to deform laterally and twist (Fig.4).The location of the critical pier,for the de finition of the target displacement,will depend on the relative deformations of the deck and arch.The fundamental mode shape determined from the modal analysis is ud here,instead of the effective mode shape procedure for multispan bridges as suggested by Kowalsky (2002)and Dw
airi and Kowalsky (2006),to determine the deformed shape of the deck and arch,and the twist of the arch at location of piers.This
fundamental
Fig.2.Fundamentals of DDBD by Priestley et al.(2007):(a)SDOF simulation;(b)effective stiffness K e ;(c)equivalent damping versus ductility;(d)design displacement spectra
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mode shape is then ud to calculate the relative deformation of the deck and arch for the assumed pinned restraint abutments in the transver direction.However,an iterative approach is likely needed to gauge the relative deformations of the deck and the arch,which will depend on the proportions of the total ba shear that they carry.The deformation capacity of the critical pier will depend on the detailing and general loading conditions.However,a chord rotation limit of 3%,as recommended in Sullivan et al.(2012),should be suitable for a repairable damage limit state.With the chord rotation capacity of the critical pier known,the displacement pro file is then t by scaling the assumed displaced shape of the deck and arch and twist of the arch,such that at least one pier reaches its limit state deformation (3%chord rotation).Thus,the final inelastic displacement pro file (of both the arch and the deck)is obtained through an iterative procedure in which the assumed displacement pro
file is ud via the DDBD procedure (Steps 2to 5)to obtain an initial design ba shear [Eq.(3)].This design ba shear is then distributed as a t of equivalent lateral forces [Eq.(5)]to a model of the structure characterized with effective stiffness properties.As explained in Steps 6and 7,the displacements obtained from the static analysis of the bridge under the t of equivalent lateral forces is compared with the initially assumed dis-placement pro file,and if the differences are negligible,the assumed displacement pro file is valid.If not,the assumed displacement pro file is updated and iterations are undertaken until convergence.
In addition to the displacement pro file,Fig.3indicates that the designer should also choo an internal force distribution,becau this will be uful for determination of the system damping in Step 3.At the initial stage of design,the internal force distribution is
not
Fig.3.Flowchart of DDBD procedure for deck arch bridges
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known.However,a t of equivalent lateral forces acting at the deck and arch levels for a unit ba shear,V b ,unit ,can be approximated,decomposing Eq.(4)to distinguish between the ismic mass at deck and arch levels,as shown
F arch,j ¼m arch,j D arch,j P n i ¼1m i i V b ,unit
tryout
F deck,k
¼m deck,k D deck,k
彩妆培训学校
P n
i ¼1m i D i
V b ,unit (5)
where F arch,j 5equivalent lateral force for unit ba shear;m arch,j 5mass;D arch,j 5displacement at location j of the arch;F deck,k 5equivalent lateral force for unit ba shear;m deck,k 5mass;and D deck,k 5displacement at location k of the deck.The denominator refers to the mass,m i ,and displacement,D i ,of point i ,with sum-mation over all n mass locations (where n 5j 1k ).
To gauge the fundamental mode proportions of ba shear,it is assumed that a percentage,x ,of the deck-level shear is taken by the abutments,and the rest of the shear force is resisted by the piers.The shear carried by the arch will then be the sum of the forces at the arch level and the shear force transferred by the piers resting on the arch.Thus,the shear force proportions (i.e.,for a unit ba shear force)taken by the abutments,piers,and arch are given by Eqs.(6)–(8),respectively
V Abt,R ¼x P r k ¼1
F deck,k
(6)
V PR ,i ¼
Q P ,i =D PR ,i
P n i ¼1Q P ,i PR ,i
!
ð12x ÞP r k ¼1F deck,k
(7)
V arch,R ¼P q j ¼1
F arch,j þ
P
Piers :on :arch
V PR ,i (8)
where D PR ,i 5(relative)displacement impod on pier i ;and Q P ,i 5relative work done of pier i and can be found from
Q P ,i ¼a P
H P ,i
D PR ,i ðductile column ÞQ P ,i
¼a P m P ,i P ,i
D PR ,i ðelastic column Þ
(9)
where a p 5flexural strength ratio (i.e.,pier strength normalized by
a reference value).The strength of the piers is a design choice and can initially be obtained by simpli fied calculations or by performing moment curvature analysis with a trial reinforcement content and an axial load calculated from static load at the ba of the pier (becau the plastic hinge forms at the pier ba).m p ,i is the ductility demand for the piers that respond elastically and should adopt values of less than 1.0to indicate the fraction of strength expected to develop in the pier.From the preceding,there may be doubts as to how to initially estimate an appropriate internal force distribution,particularly given that the proportion of lateral force carried by the abutments will depend on the degree of abutment restraint and the relative stiffness of the piers,deck superstructure,and arch.However,for the three different arch bridges examined in this paper,it was assumed ini-tially that 50%of the ba shear was carried by the abutments (i.e.,x 50:5),and only two or three iterations (Step 7)would then be required to arrive at stable final values.
Step 2:Determine Equivalent SDOF System Displacement and Effective Mass
The equivalent SDOF properties are estimated in line with the standard substitute structure approach ud in DDBD (Priestley et al.2007).The system design displacement is obtained as D d ¼P n i ¼1
m i D 2i P n i ¼1m i D i ¼P q j ¼1m arch,j D 2arch,j þP r k ¼1m deck,k D 2deck,k P q j ¼1m arch,j D arch,j þP r
k ¼1m deck,k D deck,k
(10)
Fig.4.Displaced shape of a typical deck arch bridge excited in the transver and longitudinal direction:(a)elevation of the bridge (solid lines reprent the undeformed shape,whereas dotted lines
reprent longitudinal displaced shape of the bridge);(b)ction view of transver displaced shape;(c)plan view of transver displaced shape
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