PART II IMPLEMENTATION OF THE DIRECT DISPLACEMENT- BASED DESIGN METHOD FOR SEISMIC DESIGN OF HIGHWAY BRIDGES

Transcription

1 PART II IMPLEMENTATION OF THE DIRECT DISPLACEMENT- BASED DESIGN METHOD FOR SEISMIC DESIGN OF HIGHWAY BRIDGES Vnco A. Suarez and Mervyn J. Kowalsky Department of Cvl, Constructon and Envronmental Engneerng, North Carolna State Unversty, Campus-Box 7908, Ralegh, NC-27695, USA ABSTRACT Ths paper presents the DDBD method wth all the detals requred for ts applcaton to the desgn of conventonal hghway brdges. The method presented here ncludes new features such as: the ncorporaton of the dsplacement capacty of the superstructure as a desgn parameter, the determnaton of a stablty-based target dsplacement for pers the desgn of skewed bents and abutments, among others. 1

2 1. INTRODUCTION After the Loma Preta earthquake n 1989, extensve research has been conducted to develop mproved sesmc desgn crtera for brdges, emphaszng the use of dsplacements rather than forces as a measure of earthquake demand and damage n brdges. (Prestley, 1993; ATC, 1996; Caltrans, 2006; ATC, 2003; Imbsen, 2007) Several Dsplacement Based Desgn (DBD) methodologes have been proposed. Among them, the Drect Dsplacement-Based Desgn Method (DDBD) (Prestley, 1993) has proven to be effectve for performance-based sesmc desgn of brdges, buldngs and other types of structures (Prestley et al, 2007). Specfc research on DDBD of brdges has focused on desgn of brdge pers (Kowalsky, 1995), drlled shaft bents wth sol-structure nteracton (Suarez and Kowalsky, 2007) and mult-span contnuous brdges (Calv and Kngsley, 1997; Kowalsky 2002; Dwar, 2005; Ortz, 2006). DDBD dffers from other DBD procedures for brdges, such as the Sesmc Desgn Crtera of Caltrans (2005) or the newly proposed AASHTO Gude Specfcaton for LRFD Sesmc Brdge Desgn (Imbsen, 2007), n the use of an equvalent lnearzaton approach and n ts executon. Whle the other methods are teratve and requre strength to be assumed at the begnnng of the process, DDBD drectly returns the strength requred by the structure to meet a predefned target performance. The purpose of ths report s to present the DDBD method wth all the detals requred for ts applcaton to the desgn of conventonal hghway brdges. The method presented here ncludes new features such as: the ncorporaton of the dsplacement capacty of the superstructure as a desgn parameter, the determnaton of a stablty-based target dsplacement for pers, the desgn of skewed bents and abutments, among others. 2. FUNDAMENTALS OF DDBD DDBD was frst proposed by Prestley (1993) as a tool for Performance-Based Sesmc Engneerng. The method allows desgnng a structure to meet any level of performance when subjected to any level of sesmc hazard. DDBD starts wth the defnton of a target dsplacement and returns the strength requred to meet the target dsplacement under the 2

3 desgn earthquake. The target dsplacement can be selected on the bass of materal strans, drft or dsplacement ductlty, ether of whch s correlated to a desred damage level or lmt state. For example, n the case of a brdge column, desgnng for a servceablty lmt state could mply steel strans to mnmze resdual crack wdths that requre repar or concrete compresson strans consstent wth ncpent crushng. DDBD uses an equvalent lnearzaton approach (Shbata and Sozen, 1976) by whch, a nonlnear system at maxmum response, s substtuted by an equvalent elastc system. Ths system has secant stffness, K eff, and equvalent vscous dampng, ξ eq, to match the maxmum response of the nonlnear system (Fg 1). In the case of mult degree of freedom systems, the equvalent system s a Sngle Degree of Freedom (SDOF) wth a generalzed dsplacement, Δ sys, and the effectve mass, M EFF, computed wth Eq. 1 and Eq. 2 respectvely (Fg.2) (Calv and Kngsley, 1995). In these equatons, Δ 1.Δ Δ n are the dsplacements of the pers and abutments (f present) accordng to the assumed dsplacement profle, and M 1.M.M n are effectve masses lumped at the locaton of pers and abutments (f present). Fgure 1 Equvalent lnearzaton approach used n DDBD 3

4 Fgure 2 - Equvalent sngle degree of system. Δ M sys = = n = 1 = n = 1 Δ = n = 1 eff= = n = 1 2 Δ M Δ M Δ M 2 M 2 (1) (2) The energy dsspated by nelastc behavor n the brdge s accounted for n the substtute elastc system by the addton of equvalent vscous dampng ξ eq. Wth the sesmc hazard represented by a dsplacement desgn spectrum that has been reduced to the level of dampng n the brdge, the requred effectve perod, T eff, s easly found by enterng n the dsplacement spectrum curve wth Δ sys (Fg. 3). Once T eff s known, the stffness, K eff, and requred strength, V, for the structure are computed from the well known relaton between perod, mass and stffness for SDOF systems. Fnally, V s dstrbuted among the elements that form the earthquake resstng system, and the elements are desgned and detaled followng capacty desgn prncples to avod the formaton of unwanted mechansms. 4

5 Fgure 3 Determnaton of effectve perod n DDBD 2.1 Equvalent dampng Several studes have been performed to obtan equvalent dampng models sutable for DDBD (Dwar 2005, Blandon 2005, Suarez 2006, Prestley et al 2007). These models relate equvalent dampng to dsplacement ductlty n the structure. For renforced concrete columns supported on rgd foundatons, ξ eq, s computed wth Eq. 3 (Prestley, 2007). ξ eq μ 1 t = (Eq. 3) πμ t For extended drlled shaft bents embedded n soft sols, the equvalent dampng s computed by combnaton of hysteretc dampng, ξ eq,h, and tangent stffness proportonal vscous dampng, ξ v, wth Eq. 4 (Prestley and Grant, 2005). The hysteretc dampng s computed wth Eq. 5 as a functon of the ductlty n the drlled shaft. The values of the 5

6 parameters p and q are gven n Table 1 for dfferent types of sols and boundary condtons (Suarez 2005). In Table 1, clay-20 and clay-40 refer to saturated clay sols wth shear strengths of 20 kpa and 40 kpa respectvely. Sand-30 and Sand-37 refer to saturated sand wth frcton angles of 30 and 37 degrees respectvely. A fxed head mples that the head of the extended drlled shaft dsplaces laterally wthout rotaton, causng double bendng n the element. A pnned head mples lateral dsplacement wth rotaton and sngle bendng. To use Eq. 4, ξ v should be taken as 5%, snce ths value s typcally used as default to develop desgn spectra. ξ = ξ μ + ξ eq v eq, h μ 1 (Eq. 4) ξ eq, h μ 1 = p + q μ μ 1 (Eq. 5) Table 1. Parameters for hysteretc dampng models n drlled shaft - sol systems All dampng models are plotted n Fg. 4. It s observed that when ductlty equals one, the equvalent dampng for the column on rgd foundaton equals 5% (.e. the elastc vscous dampng level) whereas the equvalent dampng for the drlled shafts s hgher than 5%. The addtonal dampng comes from the sol whch performs nelastcally and dsspates energy at dsplacements that are less than the yeld dsplacement of the renforced concrete secton. In cases where the target dsplacement s less than the yeld dsplacement of the element, a lnear relaton between dampng and ductlty s approprate. Such relaton s gven by Eq. 6. ξ = ξ + q ξ ) μ eq v ( v μ < 1 (Eq. 6) 6

7 Fgure 4. Equvalent dampng models for brdge pers 3. GENERAL DDBD PROCEDURE FOR BRIDGES The man steps of the desgn procedure are presented n Fg. 5. The brdge s prevously desgned for non-sesmc loads and the confguraton, superstructure secton and foundaton are known. A desgn objectve s proposed by defnng the expected performance and the sesmc hazard. Then, the target dsplacement profle for the brdge s determned, and DDBD s appled n the longtudnal and transverse axes of the brdge. Fnally the results are combned, P-Δ effects are checked and renforcement s desgned and detaled followng Capacty Desgn prncples (Prestley et al, 2007). 7

8 NON SEISMIC DESIGN DEFINE DESIGN OBJECTIVE DEFINITION OF TARGET DISPLACEMENT DDBD IN TRANSVERSE DIRECTION DDBD IN LONGITUDINAL DIRECTION COMBINE TRANSVERSE AND LONGITUDINAL STRENGHT DEMAND CHECK P-DELTA CAPACITY DESIGN OF RESISTING ELEMENTS Fgure 5 - DDBD man steps flowchart DDBD IN TRANSVERSE AND LONGITUDINAL DIRECTION DEFINE DISPLACEMENT PATTERN SCALE PATTERN TO REACH TARGET DISPLACEMENT COMPUTE DUCTILITY DEMAND AND EQUIVALENT DAMPING COMPUTE SYSTEM MASS, DISPLACEMENT AND EQUIVALENT DAMPING FMS BUILT 2D MODEL WITH SECANT MEMBER STIFFNESS COMPUTE INERTIAL FORCES FIND REFINED FIRST MODE SHAPE BY STATIC ANALYSIS USE FIRST MODE SHAPE AS DISPLACEMENT PATTERN DETERMINE EFFECTIVE PERIOD, STIFFNESS AND REQUIRED STRENGTH EMS FMS NO Was the dsplacement Pattern predefned? YES Is the strenght dstrbuton correct? NO EMS BUILT 2D MODEL WITH SECANT MEMBER STIFFNESS PERFORM EFFECTIVE MODE SHAPE ANALISYS YES COMPUTE DESIGN FORCES USE COMBINED MODAL SHAPE AS DISPLACEMENT PATTERN Fgure 6 - Complementary DDBD flowcharts The flowcharts n Fg. 6 show the procedure for DDBD n the transverse and longtudnal drecton, as part of the general procedure shown n Fg. 5. As seen n Fg. 6, there are three 8

9 varatons of the procedure: (1) If the dsplacement pattern s known and predefned, DDBD s appled drectly; (2) If the pattern s unknown but domnated by the frst mode of vbraton, as n the case of brdges wth ntegral or other type of strong abutment, a Frst Mode Shape (FMS) teratve algorthm s appled; (3) If the pattern s unknown but domnated by modal combnaton, an Effectve Mode Shape (EMS) teratve algorthm s appled. The drect applcaton of DDBD, when the dsplacement pattern s known, requres less effort than the applcaton of the FMS or EMS algorthms. Recent research by the author (Suarez and Kowalsky, 2008a) showed that predefned dsplacement patterns can be effectvely used for desgn of brdge frames, brdges wth seat-type of other type of weak abutments and brdges wth one or two expanson jonts. These brdges must have a balanced dstrbuton of mass and stffness, accordng to AASHTO (Ibsen, 2007). A summary of the desgn algorthms applcable to common types of hghway brdges s presented n Table 2. Table 2. Dsplacement patterns for DDDBD of brdges A detaled explanaton of all the steps nvolved n the applcaton of DDBD s presented next. 3.1 Desgn Objectve In DDBD, a desgn objectve or performance objectve s defned by specfcaton of the sesmc hazard and the desgn lmt state to be met under the specfed sesmc hazard. The sesmc hazard s represented by an elastc dsplacement desgn spectrum (Fg 3). The 9

10 desgn spectrum s characterzed by a Peak Spectral Dsplacement, PSD, and a corner perod, T c. The desgn lmt state can be based on materal stran lmts, ductlty, drft or any other damage or stablty ndex. DDBD allows any combnaton of sesmc hazard and desgn lmt state; therefore, t can be used as a tool for Performance Based Sesmc Engneerng. Some of the most common desgn lmt states are: Lfe safety Ths lmt s used n the AASHTO Gude Specfcatons for LRFD Sesmc Brdge Desgn (Imbsen, 2007) for brdges n Sesmc Desgn Category (SDC) D. It s ntended to protect human lfe durng and after a rare earthquake. Ths lmt mples that the brdge has low probablty of collapse but may suffer sgnfcant damage n pers and partal or complete replacement may be requred. To compute per dsplacements to meet the lfe safety lmt the ductlty lmts proposed n the AASHTO LRFD Gudes (Imbsen, 2007) can be used. For sngle column bents ductlty equals fve. For multple column bents, ductlty equals sx. For per walls s the weak drecton, ductlty equals fve. For per walls n the strong drecton, ductlty equals one. Damage control. Ths lmt s more restrctve than the lfe safety lmt state. It sets the lmt, beyond whch damage n pers s not longer economcally reparable due to falure of the transverse renforcement (Kowalsky, 2000). For crcular renforced concrete columns typcal strans related to ths lmt state are for concrete n compresson and 0.06 for steel n tenson. Specfc values of compresson stran for the confned concrete can be estmated usng the energy balance approach developed by Mander (1988). In ths model (Eq. 7), the damage-control concrete stran, ε c,dc, s a functon of volumetrc transverse steel rato, ρ v, yeld stress of the transverse steel renforcement, f yh, ultmate stran of the transverse steel renforcement, ε su,, compressve strength of the confned concrete, f cc, (Eq. 8) compressve strength of the unconfned concrete, f c, and the confnement stress f 1 (Eq. 9). 10

11 ρv f yhε su ε c, dc = (Eq. 7) f ' cc` 7.94 f1 f1 f ' = cc f ' c (Eq. 8) f ' c f ' c f1 = 0.5ρ v f yh (Eq. 9) The damage control dsplacement for sngle column or multple column bents can then be computed based on the damage control strans usng the plastc hnge method (Prestley and Calv, 1993). Ths method s covered n Secton Servceablty Ths lmt s more restrctve than the damage control lmt state. It sets the lmt beyond whch damage n pers needs repar (Kowalsky, 2000). For crcular renforced concrete columns, typcal strans related to ths lmt state are for concrete n compresson and for steel n tenson. The steel tenson stran s defned as the stran at whch resdual cracks wdths would exceed 1 mm. Servceablty per dsplacement can be computed wth the plastc hnge method covered n Secton Stablty lmt In addton to damage-based lmt states, a stablty crteron must also be specfed as part of the desgn objectve. Accordng to the AASHTO gude specfcaton (Imbsen, 2007), P- Δ effects can be neglected n the desgn of pers when the stablty ndex s less that 25%. For applcaton n DDBD, Prestley et al (2007) suggest that f the stablty ndex s hgher that 8%, P-Δ effects should be counteract by an ncrease n the strength of the per. However, stablty ndex should not exceed 30%. 11

12 3.2 Determnaton of target dsplacement Perhaps the most mportant step of the DDBD procedure s to determne the target dsplacement profles n the transverse and longtudnal drectons of the brdge. Ths process s executed n two steps: (1) Target dsplacements are computed for all earthquake resstng elements; (2) Target profles for transverse and longtudnal response are proposed so that one or more elements meet ther target dsplacements. No element must exceed ts target dsplacement. The proposed target profles must be consstent wth the expected dynamc response of the brdge. Therefore n most cases, t wll not be possble that all elements meet ther target dsplacements and there wll be one or two elements controllng the dsplacement profles of the brdge. A detaled explanaton on how to determne the target dsplacements for superstructures, abutments and pers, and the target dsplacement profles for a brdge s presented next Plan Curvature Ths feature cannot be explctly accounted for n DDBD. Plan curvature complcates the defnton of two prncpal desgn axes, and makes t dffcult to apply of DDBD, snce the method requres the determnaton of target dsplacement profles n each prncpal drecton ndependently. Fgure 7. Curve brdge unwrapped to be desgned as straght 12

13 A practcal soluton suggest that brdges wth subtended angles of 90 degrees or less can be unwrapped and desgned as straght brdges (Fg. 7). Ths s currently recommended n other brdge desgn codes (Caltrans, 2004; Ibsen 2007). Span lengths and skew angles n the equvalent straght brdge must be the same as n the curved brdge. Gravty nduced forces, especally those resultng from the curved geometry, must be carefully consdered and combned wth sesmc actons. (see Secton 3.8.1) Superstructure target dsplacement It s a common strategy to desgn brdges n whch damage and energy dsspaton take place n the pers and abutments whle the superstructure s protected and desgned to reman elastc. The reason s that the substructure elements can be effectvely desgned to be ductle and dsspate energy whle most superstructures cannot. Current dsplacement-based practce (Ibsen, 2007; Caltrans, 2004) focuses on a dsplacement demand-capacty check for pers and gnores the lmted dsplacement capacty of the superstructure. The mplcaton of ths s that whle pers can have enough dsplacement capacty to cope wth sesmc demand, the superstructure mght not. Therefore, a desgn based purely on per dsplacement capacty could lead to brdges n whch the superstructure suffers unaccounted damage. If a target dsplacement s determned for the superstructure, ths should meet the servceablty or other more restrctve stran lmts or should by controlled by the dsplacement capacty of expanson jonts. Ths dscusson apples only to DDBD of brdges n the transverse drecton. Brdge superstructures are usually stff and strong n ther axs, so ther performance s not an ssue when desgnng n the longtudnal drecton. Determnng a superstructure target dsplacement s mportant for desgn of brdges respondng wth a flexble transverse dsplacement pattern, ncludng: brdges wth strong abutments, brdges wth weak and flexble superstructures, brdges wth unbalanced mass and stffness. 13

14 The DDBD framework allows easy ncluson of superstructure transverse target dsplacement nto the desgn procedure. The transverse target dsplacement profle of the brdge must be defned accountng for the target dsplacement of the pers, abutments and target dsplacement of the superstructure. The determnaton of a target dsplacement for the superstructure requres moment curvature analyss of the superstructure secton and double ntegraton of the curvature profle along the length of the superstructure. Moment curvature analyss should account for expected materal propertes and strans caused by gravty loads. The result of the moment curvature analyss s the target curvature to meet the specfed stran lmts. Other mportant result s the flexural stffness of the superstructure, whch s also needed n desgn. If t s beleved that the target dsplacement of the superstructure should be controlled by yeldng of the longtudnal renforcement of a concrete deck, a yeld curvature as target curvature can be estmated wth Eq. 10. Where w s s the wdth of the concrete deck and ε y s the yeld stran of the steel renforcement. The yeld curvature of a secton n manly dependent on ts geometry and t s nsenstve to ts strength and stffness (Prestley, 2007). φ ys 2ε y = (Eq. 10) w s Fgure 8 - Assumed superstructure dsplacement pattern Assumng that the superstructure responds n the transverse drecton as a smply supported beam, wth the sesmc force actng as a unform load (Fg. 8), the curvature along the superstructure, φ s(x), as a functon of the target curvature of the superstructure, φ st, s gven by Eq. 11. Double ntegraton of the curvature functon results n the target 14

15 dsplacement functon shown as Eq. 12. Where L s s the length of the superstructure and x s the locaton of the pont of nterest. φ s( x) 4φ ysx 4φ x = (Eq. 11) L L s 2 ys 2 s x 4L + s x 2Ls x = φ Ts (Eq. 12) 6Ls ΔTs 2 Eq. 12 gves the target dsplacement relatve to the abutments. However, snce the abutments are also lkely to dsplace, Eq. 13 should be added to Eq. 12 to obtan a total target dsplacement. In Eq. 13, Δ 1 and Δ n are the dsplacements of the ntal and end abutment respectvely. Δ n Δ1 Δ = Δ + x (Eq. 13) A 1 L s x 4L s x + 2Ls x Δ n Δ1 ΔTs = φ Ts + Δ1 + x 2 (Eq. 14) 6L L s s In Eq. 14, x s replaced by the locaton of each per, x, to get the target dsplacement of the superstructure Δ Ts at that specfc pont. If Δ Ts s less than the target dsplacement of the per, then Δ Ts controls desgn and becomes the desgn target dsplacement for the per. An example that llustrates the applcaton of ths model s presented n Secton Abutment target dsplacement Due to specfc confguratons and desgn detals, an approprate estmaton of lateral target dsplacement wll n most cases requre a nonlnear statc analyss of the abutment that has been prevously desgned for non-sesmc loads. Instead of such analyss, a gross estmaton of the dsplacement that wll fully develop the strength of the fll behnd the back or wng walls can be obtaned wth Eq. 15 (Imbsen, 2008). In ths equaton, f h s a factor taken as 0.01 to 0.05 for sols rangng from dense sand to compacted clays and H w s 15

16 the heght of the wall. Ths relaton mght be useful n the assessment of a target dsplacement for ntegral abutments or seat-type abutments wth knock-off walls. Δ = f H (Eq. 15) T h w Per target dsplacement Any type of brdge per can be desgned wth DDBD as long as: (a) a target dsplacement can be easly estmated pror desgn; b) a relaton between dsplacement and equvalent dampng can be establshed. The frst requrement s usually easy to comply knowng that the relaton between stran n the plastc hnge regon and dsplacement at the top of a per s ndependent from the strength and stffness of the per. The second tem requres an assessment of desgn ductlty and the use of an exstng ductlty-dampng model such as those presented n Secton 2.1. The determnaton of ductlty demand requres knowledge of yeld dsplacement whch s calculated as part of the target dsplacement determnaton. As a result of ths, the desgn of most common types of pers used n hghway brdges can be easly mplemented n DDBD. Ths ncludes pers wth solaton/dsspaton devces and pers wth sol-structure nteracton. Table 3 - Parameters for DDBD of common types of pers PIER TYPE Effectve Heght Hp Yeld Dsp. Factor α Shear Heght Hs Transv. Long. Transv. Long. Transv. Long. Sngle column ntegral bent H+Hsup+Lsp H+2Lsp 1/3 1/6 Hp Hp/2 Sngle extended drlled shaft ntegrall bent Le+Hsup Le+Lsp vares vares Hp Hp/2 Sngle column non-ntegral bent pnned n long drecton H+Hsup+Lsp H+Lsp 1/3 1/3 Hp Hp Sngle extended-drlled-shaft non-ntegral bent pnned n long drecton Le+Hsup Le vares vares Hp Hp/2 Mult column ntegral bent H+2Lsp H+2Lsp 1/6 1/6 Hp/2 Hp/2 Mult extended-drlled-shaft ntegrall bent Le+Lsp Le+Lsp vares vares Hp/2 Hp/2 Mult column ntegral bent wth pnned base H+Lsp H+Lsp 1/3 1/3 Hp Hp Mult column bent pnned n longtudnal drecton H+2Lsp * H+Lsp * 1/6 * 1/3 * Hp/2 * Hp * Mult extended-drlled-shaft non-ntegral bent pnned n long drecton Le * Le * vares * vares * Hp/2 * Hp * * These are n-plane and out-of-plane values and must be corrected for skew to get transverse and longtudnal values 16

17 Fgure 9 - Common types of pers n hghway brdges A graphc descrpton of common types of renforced concrete pers used n conventonal brdges s presented n Fg. 9. In relaton to that fgure, Table 3 shows values for: the effectve heght of the per H p, yeld dsplacement factor α, and shear heght H s. 17

18 Fgure 10 - Le and a for defnton of the equvalent model for drlled shaft bents (Suarez and Kowalsky, 2007) These parameters are gven for desgn n transverse and longtudnal axes of the brdge. In Table 3, H s the heght of column, L e s the effectve heght of drlled shaft, H sup s the heght from the sofft to the centrod of the superstructure and L sp s the stran penetraton length. The parameters L e and α for drlled shaft bents are shown n Fg. 10 n terms of the type of sol, above ground heght of the bent, L a, and dameter of the drlled shaft secton D. Plastc Hnge Method Stran-based target dsplacements are determned usng the plastc hnge method (Prestley et al, 2007). For any type of per lsted n Table 3, the target dsplacement, Δ t, along the 18

19 transverse and longtudnal axs of the brdge s estmated wth Eq. 16. In ths equaton, Δ y s the yeld dsplacement of the per, φ t and φ y are the target and yeld curvature respectvely, L p s the plastc hnge length and H p s the effectve heght of the per defned n Table 3. Δ T = Δ y + ( φ φ ) L H (Eq. 16) t y p p The target curvature s determned n terms of the target concrete stran, ε c,t, target steel stran, ε s,t, and neutral axs depth, c, wth Eq. 17. The target curvature can be controlled by the concrete reachng ts target stran n compresson or the flexural renforcement reachng ts target stran n tenson. ε c, ε T s, T φ dc = mn, (Eq. 17) c D c The neutral axs depth can be estmated wth Eq. 18, where P s the axal load actng on the element and A g s the gross area of the secton (Prestley et al, 2007) c P 0.2D (Eq. 18) f ' ce A = g The yeld curvature φ y s ndependent of the strength of the secton and can be determned n terms of the yeld stran of the flexural renforcement ε y and the dameter of the secton D wth n Eq. 19 (Prestley et al, 2007) ε y φy = D (Eq. 19) The yeld dsplacement Δ y s gven by Eq. 20, where α s gven n Table 3 for transverse and longtudnal drectons. For extended drlled shaft bents, α s shown n Fgure 10. y y ( H ) 2 Δ = αφ (Eq. 20) p 19

20 Lfe safety target dsplacement Is computed ntroducng the ductlty lmts gven n Secton 3.1 n Eq. 21. Δ T = Δ y μ (Eq. 21) Damage control target dsplacement The damage control concrete compresson stran ε c,dc must be frst computed wth Eq. 7. The damage control tenson stran for the flexural renforcement s ε s,dc = 0.06 (Kowalsky, 2000) (See Secton 3.1). These values are used n Eq. 17 to fnd φ t and fnally Eq. 16 s used to get the target dsplacement. Servceablty target dsplacement The stran lmts n Secton 3.1 are used to compute a target curvature and then a target dsplacement wth Eq. 16 Target dsplacements for SDC B and C For these SDCs, the equatons gven n AASHTO (Ibsen, 2007) for the assessment of dsplacement capacty for moderate plastc acton (Eq. 23) and mnmal plastc acton (Eq. 24) can also be used n DDBD to get a target dsplacement. In Eq. 23 and Eq. 24, H c s the clear heght of the columns and Λ s 1 for columns n sngle bendng and 2 for columns n double bendng. D Δ c = 0.003H c 2.32 ln 1.22 Λ H c H (m) (Eq. 23) c D Δ c = 0.003H c 1.27 ln 0.32 Λ H c H (m) (Eq. 24) c Stablty-based target dsplacement 20

21 A target dsplacement for a brdge per to meet a predefned value of the stablty ndex, θ s, can be estmated wth Eq. 25 (Suarez and Kowalsky 2008b), where the parameters a, b, c and d are gven n Tables 4 and 5 for pers on rgd foundatons and extended drlled shaft bents n dfferent types of sols. Table 4 gves the parameters for near fault stes and Table 5 for far fault stes. The parameter C n Eq. 25 s computed wth Eq. 26 n terms of the peak spectral dsplacement, PSD, the corner perod, T c, the axal load actng on the per, P, the effectve mass on the per, M eff, and the heght of the per H. C d μ θs = a + bc + c (Eq. 25) C TcΔ y P C = (Eq. 26) 2πPSD θ M H s eff If a brdge per s desgned as a stand-alone structure, M eff can be computed takng a trbutary area of superstructure and addng the mass of the cap-beam and a porton of the mass of the per tself (1/3 s approprate). If the target dsplacement s beng determned for a per that s part of a contnuous brdge, M eff s computed wth Eq. 27 as a fracton of the effectve mass of the brdge, M EFF. The effectve mass of the brdge s computed as wth Eq. 2 and v s computed wth Eq.31 M = v M (Eq. 27) eff EFF Table 4 - Parameters to defne Eq.15 for far fault stes Table 5 - Parameters to defne Eq.15 for near fault stes 21

22 The target ductlty obtaned from Eq. 25 s multpled to the yeld dsplacement of the per to get the stablty-based target dsplacement. The target ductlty obtaned from Eq. 25 s plotted n Fgs 11 and 12 for a range of values of C. In both fgures t s observed that ductlty s very senstve to changes n C when C s less than 0.5. The dfferences n ductlty for the dfferent models come from the dfferences n equvalent dampng (see Secton 2.1). If two pers have the same value of C, the one wth more dampng develops less ductlty to reach the stablty lmt. The reason s that the per wth less dampng requres more strength. The dfferent dampng models have less effect n the relaton between μ θs and C for near fault stes. Fgure 11. Ductlty vs. C for far fault stes The use of Eq. 25 avods the need of teraton n DDBD for desgns controlled by P-Δ effects. The proposed model s accurate for desgn of pers as stand-alone structures or n the case of regular brdges wth a balanced dstrbuton of mass and stffness. If the model s used wth rregular brdges, t wll produce conservatve estmates of target dsplacement 22

23 (Suarez and Kowalsky, 2008b). An example that llustrates the applcaton of ths model s presented n Secton Ductlty c Rgd Base Clay 20 Pnned Clay 20 Fxed Clay 40 Pnned Clay 40 Fxed Sand 30 Pnned Sand 30 Fxed Sand 37 Pnned Sand 37 Fxed Fgure 12. Ductlty vs. C for near fault stes 3.3 Skewed pers or abutments From a desgn perspectve, the effect of a skewed confguraton s that n-plane and out-ofplane response parameters of abutments and pers are no longer orented n the prncpal desgn drectons of the brdge (Fg. 12). (a) Skewed bent (b) Element and brdge desgn axs Fgure 12 - Desgn axes n skewed elements 23

24 The effects of skew can be consdered n DDBD by determnng the projecton of any response parameter such as yeld dsplacement Δ y, target dsplacement Δ T, shear heght H s and others, wth respect to the transverse and longtudnal drecton of the brdge. Such determnaton can be done usng an ellptcal nteracton functon between the n-plane and out-of-plane response, as gven by the followng equatons: rp rp T L rpout rpin = rpin + skew (Eq. 28) 90 rpin rpout = rpout + skew (Eq. 29) 90 Where, rp IN s the value of the response parameter n the n-plane drecton of the element, rp OUT s the response parameter n the out-of-plane drecton of the element, rp T s the projecton of the response parameter n the transverse drecton of the brdge and rp L s the projecton of the response parameter n the longtudnal drecton of the brdge. An example that llustrates the applcaton of ths model s presented n Secton Determnaton of target dsplacement profles Once the target dsplacement has been determned for the superstructure, abutments and pers, consderng skew f necessary, the next step s to propose a target dsplacement profle along the transverse and longtudnal drectons of the brdge Transverse dsplacement profles As was mentoned before, much of the effort n the applcaton of DDBD for transverse desgn of brdges s n the selecton or determnaton of the target dsplacement profle. Prevous research conducted by Dwar and Kowalsky (2006) focused on the dentfcaton of dsplacement patterns for symmetrc and asymmetrc brdge confguratons usng Inelastc Tme Hstory Analyss (ITHA). Three types of dsplacement patterns were dentfed, namely: Rgd Body Translaton (RBT), Rgd Body Translaton wth Rotaton (RBTR) and flexble pattern. These patterns were found to be hghly dependent on the 24

25 superstructure to substructure stffness rato, brdge regularty and type of abutment. It was also found that only symmetrcal regular brdges wth stff superstructures and free abutments respond wth a RBT pattern. Recent research (Suarez and Kowalsky, 2008) has shown that brdges frames and brdges wth weak abutments that comply wth balanced mass and stffness requrements of the AASHTO gude specfcaton (Imbsen, 2007) can be desgned n the transverse drecton assumng a rgd body translaton pattern. The Balanced Mass and Stffness (BMS) ndex s computed wth Eq. 30, where m and m j are the lumped mass at pers and j and K p and K pj are the stffness of pers and j. Ths ndex s computed n two ways, BMS1 s the least value that results of all combnatons of any two pers and BMS2 s the least value that results from all combnatons of adjacent pers. If BMS1 > 0.50 and BMS2 > 0.75 the brdge s consdered regular. M K p j BMS = (Eq. 30) M K j p AASHTO (Imbsen, 2007) and Caltrans (1996) recommend brdges to comply wth the balanced mass and stffness requrements descrbed above. Ths s to avod torson and uneven dstrbuton of damage n the structures. Therefore, t s lkely that most conventonal hghway brdges can be desgned wth DDBD usng a rgd body dsplacement pattern. In such a case, t s expected that the transverse dsplacement of all pers be the same and thereof, the ampltude of dsplacement profle s controlled by the per or abutment wth the least target dsplacement (See Table 2). When a rgd body dsplacement pattern s used, the applcaton of DDBD s drect as shown n Fg. 6 Ths process s cover n detal n Secton In the case of brdges wth ntegral abutments or other types of strong abutments, desgned to lmt the dsplacement of the superstructure, the transverse dsplacement pattern s flexble and ts shape depends manly on the relaton between the stffness of the superstructure, abutments and pers (See Table 2). Snce the stffness of the pers and abutments depends on ther strength, whch s not know at the begnnng of the process, the shape of the dsplacement pattern of ths type of brdges must be assumed and refned 25

26 teratvely. Ths s done usng the Frst Mode Shape (FMS) or the Effectve Mode Shape (EMS) algorthms shown n Fg. 6 and descrbed n detal n Sectons and The ampltude of the dsplacement profles s set so that no element exceeds ther target dsplacement. For brdges wth up to two nternal expanson jonts n the superstructure, research by the authors (Suarez and Kowalsky, 2008a) ndcated that predefned lnear dsplacement patterns can be used to execute DDBD drectly. Such dsplacement patterns are shown n Table 2 and Fg. 13, the ampltude of the dsplacement profles s set so that no element exceeds ther target dsplacement. EXP. JOINT 1=0 n=0 1=0 n=0 (a) Elevaton vew (b) dsplacement pattern Fgure 13 - Dsplacement patterns for brdges wth expanson jonts Longtudnal dsplacement profles Contnuous superstructures are usually strong and axally rgd, therefore t s reasonable to assume that the superstructure dsplaces longtudnally as a rgd body, constranng the dsplacements of all pers to be the same. In ths case, the ampltude of the dsplacement profle s controlled by the per or abutment wth the least target dsplacement and DDBD s appled drectly as shown n Fg

27 For superstructures wth expanson jonts, two possble dsplacement patterns must be consdered (Fg. 14), one wth the expanson jont gaps closng towards one end of the brdge and the other wth the jonts closng towards the opposte end of the brdge. These two alternatve patterns are lkely to result n dfferent generalzed dsplacement for the brdge (See Secton 2, Eq. 1). The least generalzed dsplacement controls desgn. Fgure 14 Longtudnal dsplacement pattern for brdges wth expanson jonts 3.5 DDBD IN THE TRANSVERSE DIRECTION DDBD s appled n the transverse drecton as part of the overall procedure showed n Fg. 5. A summary of the steps nvolved n the transverse desgn s presented n Fg. 6. In ths fgure t s shown that there are three dfferent desgn algorthms. A detaled descrpton of the general steps and algorthms s presented next. Strength Dstrbuton If a brdge s desgned to suffer damage durng the desgn earthquake, t s possble that most of ts pers and abutments (f present) develop ther nomnal strength and behave nelastcally. Therefore, the dstrbuton of strength among bents and abutments s not longer a unque matter of ntal stffness but a desgner s choce constraned by equlbrum. 27

28 Fgure 15 Strength dstrbuton n the longtudnal and transverse drectons In most brdges, there are two load paths for the nertal forces developed durng an earthquake, one s from the superstructure through the abutments and the other s from the superstructure through the pers to the foundaton sol (Fg. 15). The rato of total base shear V taken by the superstructure to the abutments v s s generally unknown and must be assumed at the begnnng of the desgn process. The rato of base shear taken by the pers, v p =1-v s, must be dstrbuted among them satsfyng equlbrum. Some of the possble dstrbutons of strength are: A dstrbuton to obtan per columns wth the same flexural renforcement rato, whch s practcal and leads to constructon savngs. A dstrbuton wth equal shear demand n all pers, whch s approprate for brdges wth sesmc solaton, to use the same devce n all pers Strength Dstrbuton for columns wth same renforcement rato Snce renforced concrete sectons wth the same renforcng rato show smlar ratos of cracked to gross secton stffness, a dstrbuton of strength to get the same renforcement rato n all columns of the brdge can be computed wth Eq. 31. Ths equaton gves the rato of total strength, v, taken by bent, wth n columns of dameter D, shear heght Hs and ductlty μ, requred to satsfy force equlbrum. Table 3 defnes H s for dfferent types of pers. 28

29 = ( 1 vs) nμd H s v 3 nμd H 3 s μ 1 (Eq. 31) Strength Dstrbuton for equal shear n bents Strength among pers can be dstrbuted equally. Ths alternatve s practcal when desgnng brdges wth solaton/dsspaton devces on top of the bents. The reason s that the same devce can be use n all brdge pers Drect procedure Ths procedure apples to cases where a rgd body or other predefned dsplacement profle has been adopted for desgn (see Secton 3). Step1. Determne Target dsplacement profle Follow Secton 3.4 Step2. Determne propertes of equvalent SDOF system Usng Eq. 1 and Eq. 2 the generalzed dsplacement Δ sys and the Effectve mass M EFF are found. Step3. Compute ductlty and equvalent dampng For each per and abutment the ductlty s computed as the rato of the dsplacement n the target profle Δ and the yeld dsplacement Δ y of the element. If the yeld dsplacement Δ y was not computed durng the determnaton of target dsplacement, t should be computed wth Eq. 20, wth reference to Table 3. Once ductlty s known, the equvalent dampng for each per s computed wth an approprate model. See Secton 2.1. For the abutments t s recommended to use equvalent dampng values between 5% and 10%. The hgher value of dampng s approprate when full sol moblzaton s expected. Hgher values of dampng can be computed wth Eq. 3 when plastc hnges are expected to develop n ples supportng the abutment at the target 29

30 dsplacement level. If elastc response s expected n the superstructure, dampng between 2% and 5% should be used. The equvalent dampng computed for each element must be combned to obtan the equvalent dampng for the substtute SDOF system. The combnaton s done n terms of the work done by each element (Kowalsky, 2002). The work done by the superstructure s computed wth Eq. 32. The work done by each abutment s computed wth Eq. 33 and Eq. 34. The work done by each per s computed wth Eq. 35. Fnally the combned dampng s computed wth Eq. 36. ws = Δ sys ( Δ1 + Δ 2 nb ) v s (Eq. 32) wa 1 Δ1v1 = (Eq. 33) wa wp n = Δ v (Eq. 34) n n = Δ v (Eq. 35) n 1 wa1ξ1 + wanξ n + wsξ s + wpξ 2 ξ sys = n 1 (Eq. 36) wa + wa + ws + wp 1 n 2 Step 4. Determne the requred strength Frst the 5% dampng elastc dsplacement spectrum s reduced to the level of dampng of the structure (ξ sys ). To do ths, a demand reducton factor s computed wth Eq. 37 (Eurocode, 1998), where α = 0.25 for near fault stes and α = 0.5 for other stes. Then, the reduced dsplacement desgn spectra s entered wth Δ sys to fnd the requred perod, T eff, for the structure. Ths s shown schematcally n Fg. 3. Ths process has been syntheszed n Eq. 38. R ξ 7 = 2 + ξ sys α (Eq. 37) 30

31 T eff Δ sys = Tc (Eq. 38) PSD R ξ Based on the well know relaton between stffness, mass and perod of a sngle degree of freedom system, the stffness, K eff, and then the strength, V, requred for the equvalent system are obtaned wth Eq. 39 and Eq. 40 respectvely. K eff 4π M = (Eq. 39) T 2 eff 2 eff V = K eff Δ sys (Eq. 40) Step 5. Dstrbute requred strength The requred strength V s determned for each substructure element wth Eq. 41. Fnally, the desgn flexural strength, M, for the columns n each per s computed accordng to the type of per (Table 3) wth Eq. 42. V = vv (Eq. 41) H V s M = (Eq. 42) n Frst mode shape algorthm Ths algorthm s applcable to cases where a flexble dsplacement pattern has been adopted and the response of the brdge s controlled by ts frst mode of vbraton (See Table 3). Ths procedure s teratve and use elastc analyss of refne the assumed dsplacement pattern. Ths algorthm converges to the frst mode nelastc shape of the structure. Step1. Determne Target dsplacement profle Follow Secton 3.4 Step 2. Apply drect procedure 31

32 Conduct all the steps 2-4 of the drect soluton presented n Secton Step 3. Fnd a best estmate of the dsplacement pattern Inertal forces consstent wth the assumed dsplacement pattern are appled to a brdge model wth stffness secant to maxmum response. Ths statc analyss results n a dsplacement pattern that s a best approxmaton to the frst mode shape of the brdge. To perform the statc analyss, a 2D model of the brdge s bult (Fg. 16). In ths model, the substructure elements are sprngs wth secant stffness computed wth Eq.43. The superstructure s modeled as a seres of frame elements. The loads apply to the model are the nertal forces, F, are computed wth Eq. 44. K eff V Δ = (Eq. 43) F mδ = n mδ 1 V (Eq. 44) Once performed, the outcome of ths analyss ncludes the dsplacements for each substructure element, Φ, and new estmates of the proporton of the total shear taken by the abutments, v r1 and v rn. These values are computed wth Eq. 45 and Eq. 46. If v 1 and v n where assumed at the begnnng of the process, an error between the new and assumed values can be computed wth Eq. 47 and Eq. 48. Φ1K = (Eq. 45) V 1 v r 1 v r n Φ n K n = (Eq. 46) V v1 vr1 ERR = v1 abs vr1 (Eq. 47) vn vrn ERR = vn abs vrn (Eq. 48) 32

33 Fgure 16 2D model of brdge wth secant stffness. Step 4. Scalng the frst mode shape to meet the desgn objectve Ths process must assure that no element n the brdge exceeds ther prevously calculated target dsplacement. Ths step s accomplshed by calculatng a scalng factor, Sf, for each element as shown n Eq. 49. Then, the least of the scalng factors s appled to the dsplacements computed n Step 3 to get a refned dsplacement profle (Eq. 50). Sf Δ Φ T = (Eq. 49) r n mn 1 Sf Δ = Φ (Eq. 50) If the abutments are elastc, ther target dsplacements must be enforced n the dsplaced shape, that s Δ r1 = Δ y1 and Δ rn =Δ yn. An error between the prevous dsplacement profle and the new dsplacement profle can be estmated wth Eq. 51 ERR dp = = n = 1 2 ( Δ Δ ) (Eq. 51) r Step 5. Check convergence and terate When applyng ths algorthm two assumptons are made: (1) The shape of the dsplacement pattern, (2) The proporton of total shear carred by the abutments (f present). If errors ER v1 and ER vn and ER dp are greater than a predefned tolerance, the procedure must be repeated from Step 2 the number of tmes requred to get the errors 33

34 wthn the tolerance. In each teraton the proporton of base shear taken by the abutments and the dsplacement profle must be updated (.e. v 1 = v r1, v n = v rn and Δ = Δ r ) Effectve mode shape algorthm Ths algorthm s applcable to all cases (See Table 3), however due to ts ncreased complexty, t s recommended to cases where a flexble dsplacement pattern s used and the response of the brdge s controlled by the combnaton of several modes of vbraton. Ths procedure s more complex and requres more effort than the Frst Mode Shape algorthm. However, ths procedure captures the effects of hgher modes of vbraton n the dsplacement profle and n the element forces. Ths procedure s teratve and uses response spectrum analyss to refne the assumed dsplacement pattern. Its applcaton follows the steps presented next: Step1. Determne Target dsplacement profle Follow Secton 3.4 Step 2. Apply drect procedure Conduct all the steps 2-4 of the drect procedure presented n Secton Step 3. Fnd a best estmate of the dsplacement pattern A response spectrum analyss s conducted wth a model of the brdge n whch the substructure elements have secant stffness computed wth Eq. 43. A composte acceleraton response spectrum s used such as the one shown n Fg. 17. In the composte spectrum, the spectral ordnates correspondng to the effectve perod (T eff ) of the brdge are reduced by the factor R ξ computed wth Eq. 38 n Step 4 of the drect procedure. The spectral ordnates for perods shorter than T eff are not reduced snce the hgher modes are lkely to be elastc modes of response n the structure. 34

35 Fgure 17 Composte acceleraton response spectrum The outcome of ths analyss ncludes the dsplacements for each substructure element, Φ, new estmates of the proporton of the total shear taken by the abutments, v r1 and v rn, and forces n the superstructure and abutments. Forces n nelastc pers are not consdered snce any demand n excess of the capacty of the pers s only a result of the elastc analyss. The proporton of shear taken by the abutments s computed wth Eq. 45 and Eq. 46. If v 1 and v n where assumed at the begnnng of the process, an error between the new and assumed values can be computed wth Eq. 47 and Eq. 48. Step 4. Scalng the frst mode shape to meet the desgn objectve Ths process must assure that no element n the brdge exceeds ther prevously calculated target dsplacement. Ths step s accomplshed by calculatng a scalng factor, Sf, for each element as shown n Eq. 49. Then the least of the scalng factors s appled to the dsplacements computed n Step 3, to get a refned dsplacement profle (Eq. 50). 35

36 If the abutments are elastc, ther target dsplacements must be enforced n the dsplaced shape, that s Δ r1 = Δ y1 and Δ rn =Δ yn. An error between the prevous dsplacement profle and the new dsplacement profle can be estmated wth Eq. 51 Step 5. Check convergence and terate When applyng ths algorthm to two assumptons basc are made: (1) The shape of the dsplacement pattern, (2) The proporton of total shear carred by the abutments. If errors ER v1 and ER vn and ER dp are greater than a predefned tolerance, the procedure must be repeated from Step 2 the number of tmes requred to get the errors wthn the tolerance. In each teraton the proporton of base shear taken by the abutments and the dsplacement profle must be updated (.e. v 1 = v r1, v n = v rn and Δ = Δ r ) 3.6 DDBD n the longtudnal drecton Applyng DDBD n the longtudnal drecton of the brdge s generally smpler than dong t n the transverse drecton. The steps nvolved are detaled next: Step1. Determnng target dsplacement profle Due to hgh axal stffness of the superstructure, a target dsplacement profle can be determned at the begnnng of the process. There are two possbltes that must be consdered: a) If there are not expanson jonts n the superstructure nor at the abutments (f exst), all substructure elements are constraned to dsplace the same amount. Therefore, the substructure element wth the least target dsplacement wll control the target dsplacement of the brdge. b) If there are expanson jonts n the brdge, two target dsplacement profles must be consdered, one wth the brdge beng pushed towards the end per or abutment and the other wth the brdge beng pushed towards the ntal per or abutment (Fg. 14) In any case then target profle must be scaled keepng n mnd that no substructure element should exceed ts target dsplacement. Step 2. Computng system dsplacement and mass 36

37 The same as step 2 of the procedure for DDBD n the transverse drecton usng the drect procedure. Step 3. Computng ductlty demand and equvalent dampng The same as step 3 of the procedure for DDBD n the transverse drecton usng the drect procedure. Step 4. Determnng the effectve perod, secant stffness and requred strength The same as step 4 of the procedure for DDBD n the transverse drecton usng the drect procedure. Step 5. Checkng the assumed strength dstrbuton If the brdge has abutments wth known strength, the assumed values of v 1 and v n, must be verfed. New refned values for v r1 and v rn can be computed wth Eq. 52 and Eq. 53. An error between the new and assumed values can be computed wth Eq. 47 and Eq. 48. v v r1 r1 K eff,1 = (Eq. 52) K eff K eff, n = (Eq. 53) K eff If errors ER v1 and ER vn and ER dp are greater than a predefned tolerance, the procedure must be repeated from Step 2 the number of tmes requred to get the errors wthn the tolerance. In each teraton the proporton of base shear taken by the abutments must be updated (.e. v 1 = v r1, v n = v rn ) 3.7 Concurrent orthogonal exctatons Real earthquakes nduce three dmensonal dsplacements wth two horzontal components that could cause damage n the earthquake resstng components of the brdge. For ths reason, DDBD must be appled n the two prncpal drectons of the brdge and the results must be combned to produce a desgn capable of performng satsfactorly when attacked 37

38 n any drecton. Strength demands along prncpal drectons can be combned usng the well known 100%-30% rule, currently used most sesmc desgn codes (Imbsen, 2007; Caltrans, 2006 ). Applcaton of ths rule s presented n the followng secton. 3.8 Element Desgn Per desgn The desgn of pers s based n Capacty Desgn prncples to assure that all nelastc behavor take place n well detaled sectons. The flexural desgn of renforced concrete crcular per columns requres knowledge of moment, axal force and desgn concrete stran n the crtcal secton of the element. The moments n the transverse, M t, and longtudnal drecton, M l, are gven by Eq. 54 and Eq. 55, n terms of the desgn shear that resulted of the applcaton of DDBD, the shear heght gven by Table 3 and the number of columns n the per, n c. V H t st M t = (Eq. 54) nc V H l sl M l = (Eq. 55) nc The 100%-30% rule for combnaton of forces caused by concurrent orthogonal combnatons can be used to determne the desgn moment, M E. In whch case M E s the grater of the two combnatons of transverse M t and longtudnal M l moment demands gven by Eq. 56 or Eq = Mt 0.3Ml (Eq. 56) M E = Ml 0.3Mt (Eq. 57) M E + The axal load P G comes from a dead load analyss. The desgn stran ε D must be reached when M E s developed n the secton. When desgnng for mnmal or moderate nelastc actons, ε D can be taken as for nomnal strength. When desgnng for the damage- 38

39 control or lfe-safety lmt states, ε D s taken as the least of the strans calculated n the crtcal secton at the transverse or longtudnal desgn dsplacement. The stran ε D s gven by Eq. 58. Δ Δ y ε D = + φ c > y (Eq. 58) L p H p Once P G, M E and ε D are known, the amount of flexural renforcement s determned by secton analyss. Ths analyss should be based on realstc materal models and expected rather than specfed materal propertes. In practce, a moment-curvature analyss program could be used to determne by teraton the amount of flexural renforcement requred. The shear desgn s based on a demand capacty analyss. The shear demand s based on the flexural over-strength of the secton and must be computed n the desgn drecton wth the least shear heght (H s ). The shear demand s gven by Eq. 59, where Φ s an over-strength factor and H s s the shear heght gven n Table 3 for dfferent types of pers. If stran hardenng was consdered n the flexural desgn, Φ = 1.25 s approprate (Prestley et al, 2007). The shear capacty of the secton can be determned usng the modfed UCSD model (Kowalsky and Prestley, 2000). V D ΦM H E = (Eq. 59) s Combnaton of sesmc and non-sesmc loads Element desgn n DDBD ams to provde the requred strength to the earthquake resstng system so the desgn objectve s met. The resultng structure should be capable of resstng gravty loads, other non-earthquake actons and the desgn earthquake attackng n any drecton. Desgn for non-earthquake loads should follow LRFD practce specfed by AASHTO (2004). When desgnng pers n DDBD, t s not generally necessary to combne sesmc wth non-sesmc moment demand. It s also not generally necessary to consder the axal loads generated by the earthquake. The reason s that combnng sesmc wth non-sesmc 39

40 moment and accountng for earthquake nduced axal forces mght lead to unnecessary over-strength n the structure. For example, f the columns n the bent shown n Fg. 18a are only desgned for gravty axal loads P G and earthquake moments M E, the strength developed by the bent n a Pushover analyss (Fg. 18b) wll match the desgn strength V. If the sesmc nduced axal loads P E are consdered, the wndward column, wth less axal load, requres more renforcement. However due to sesmc reversal, the two columns must be provded wth the same renforcement, causng the bent to have more lateral strength than requred, as shown n the pushover curve n Fg. 18b. If gravty load moments are ncluded, the moment demand on the leeward column s ncreased, ncreasng also the amount of requred renforcement. Due to sesmc reversal, the two columns should be detaled wth the same amount of renforcement, although only one really needs that much strength. The excess of renforcement n one column, as n the prevous case, causes unnecessary overstrength. Fgure 18 Force-dsplacement response of bents desgned for combnatons of sesmc and non-sesmc forces. 40

41 Lateral over-strength s beleved to be unnecessary unless t s requred to accommodate non-sesmc loads. For the example presented prevously, f the columns are desgned for M E and P G only, after the unavodable gravty loads act, the lateral capacty of one column s reduced whle the capacty of the other column s ncreased. Under lateral loadng, one column mght experence more damage than the other; however the overall response of the bent wll be as planned. In the other hand, f the bent has over-strength n an attempt to lmt the damage n a crtcal secton of the bent, due to the over-strength, the bent wll have a conservatve response and not even the crtcal secton wll reach the desgn lmt sate, whch s thought to be uneconomcal. The arguments presented above apply to all desgn stuatons where the gravty loads produce moments that ncrease the lateral capacty of some columns and reduce the capacty of others. No consderaton of gravty moments and sesmc nduced axal forces s especally approprate for cases where desgn s controlled by P-Δ effects rather than damage-based lmt states. Cauton must be exercsed n the case of curved brdge frames wth ntegral sngle column pers, where gravty loads reduce the lateral capacty of all columns. Sesmc axal loads should not be gnored n columns wth hgh axal load ratos, where addtonal axal loads could lead to brttle falures Abutment desgn The effects that the abutments have n the response of the brdge depend on ther contrbuton of strength and stffness. Seat-type abutments have sacrfcal shear keys and knock off walls and generally contrbute lttle to the strength of the brdge. Integral abutments are monolthc wth the superstructure and must be strong enough to restran superstructure dsplacement and avod excessve damage. DDBD can be used to obtan the desgn actons for these types of abutments, as explaned next: Integral Abutments Generally, abutments have less dsplacement capacty than the adjacent bents. Therefore, ntegral abutments must be effectve n restranng the dsplacement of the superstructure at 41

42 ther ends to avod excessve damage. To do ths, ntegral abutments must be strong and stff. To desgn ntegral abutments wth DDBD, the desgner specfes, a maxmum dsplacement to be reached by the abutments n the n-plane and out-of-plane drectons. The desgner must also specfy the amount of vscous dampng and a skew angle f any. The amount of dampng should be between 5 and 10%, as recommended by AASHTO (Imbsen, 2007) for contnuous brdges, n whch there s consderable sol moblzaton at the abutments. If plastc hnges are expected to develop n the supportng ples, the amount of vscous dampng could be further ncreased as a functon of ductlty demand (See secton 2.1). Contnuous brdges wth ntegral abutments are expected to respond n the transverse drecton wth a flexble dsplacement pattern. The shape of the pattern s unknown before desgn and s generally gven by the frst mode of vbraton. Therefore DDBD s appled wth the FMS algorthm or the EMS algorthm (Table. 2) (see Secton 3.5). If the target dsplacement s less than ther yeld dsplacement, the abutments wll perform elastcally durng the earthquake. If that s the case, t s mportant to consder the addtonal demands generated by hgher modes of vbraton. Such determnaton can be accomplshed usng the EMS algorthm or by the applcaton of a dynamc amplfcaton factors. Analyses performed by the authors on a set of ten brdges wth ntegral abutments, showed that the shear demand n the abutments can be ncrease up to 2.5 tmes the forces derved from the FMS desgn (Suarez and Kowalsky, 2008a) The applcaton of DDBD n the longtudnal drecton s drect for brdges wth ntegral type of abutments. The rato of the total base shear taken by the abutments can be decded by the desgner at the begnnng of desgn. Seat type abutments 42

43 Ths type of abutments are typcally desgned for non-sesmc loads and protected from sesmc demands by the provson of sacrfcal shear keys and knock off walls that are desgned to fuse. In the n-plane drecton, contrbuton to the strength of the brdge s gven by passve pressure of the sol that s moblzed behnd the wng walls and resdual strength or frcton after the shear keys fal. In the longtudnal drecton, the strength contrbuton comes from passve pressure of the sol moblzed behnd the back-walls that s compressed. Methods for determnng the strength and yeld dsplacement of these elements are well known and presented n detal n AASHTO Gude Specfcaton for LRFD Sesmc Brdge Desgn (Ibsen, 2007). When desgnng a contnuous brdge wth seat-type abutments, ther contrbuton to the strength and the effect on the response of the brdge can be easly accounted for n DDBD. The force-dsplacement response of the abutment must be known n both drecton of desgn. In ths case, DDBD wll lmt the response of the abutment to the specfed target dsplacement by allocatng the approprate level of strength n the brdge pers. Snce the rato of the total base shear taken by the abutments s not known at the begnnng desgn, DDBD requres teraton. The study of hgher mode effects s unnecessary for elasto-plastc abutments snce demand cannot exceed capacty. 4. COMPUTERIZATION OF DDBD The computer program DDBD-Brdge has been developed for automaton of the DDBD method for hghway brdges. In most desgn cases DDBD can be appled wth manual or spreadsheet calculatons usng the drect procedure. However, tme n the applcaton of the FMS or EMS algorthms and secton desgn could be saved by programmng the algorthms nto a computer code. DDBD-Brdge has been programmed followng the general procedure presented n ths report. The program has the followng features: DDBD of hghway brdges n the transverse and longtudnal drectons Desgn usng the drect, FMS and EMS algorthms 43

44 Contnuous superstructures or superstructures wth expanson jonts Integral or seat-type abutments All types of pers shown n Table 3 Automated secton desgn by moment-curvature analyss The program and ts documentaton can be accessed and used on-lne though the Vrtual Laboratory for Earthquake Engneerng ( (Dr. Kowalsky, ths URL wll be avalable startng on July 8) 4.1 DDBD usng commercally avalable structural analyss software Any frame analyss program capable of performng statc and response spectrum analyss can be used to perform the Step 3 of the FMS or EMS algorthms presented n Secton 3.5. Even though statc analyss s not requred n the drect soluton (Secton ), nerta forces can be calculated (Eq. 44) and appled to an elastc model of the brdge to get element forces and easy element desgn. Ths s equvalent to apply the FMS algorthm (Secton 3.5.2) wthout teraton. When applyng the teratve desgn algorthms, the secant stffness of the elements must be updated n the model n each teraton. If a 3D model s used wth the pers modeled usng frame elements, Eq. 60 can be used to compute nerta reducton factors, I rf, for each element. In ths equaton, E s the elastc modulus used n the analyss program, I g s the gross moment of nerta of the secton and γ s 1/3 for pnned head columns and 1/12 for fxed head columns. Inerta reducton factors are accepted n most analyss programs to modfy the gross secton propertes of frame elements. I rf K H 3 = (Eq. 60) γei g 44

45 4.1 Evaluaton of DDBD wth Inelastc Tme Hstory Analyss Inelastc Tme Hstory Analyss (ITHA) s consdered to be the most accurate tool to assess structural performance. Other assessment methods based on nonlnear statc analyss (Pushover) such as the Capacty Spectrum Method (FEMA, 2005) or the Dsplacement Coeffcent Method (FEMA, 2005) should not be used for assessment of structures desgned wth DDBD snce these methods also use equvalent lnearzaton and an assumed pattern of dsplacement. Therefore, there s no reason to beleve that the performance predcted by assessment methods based on Pushover analyss s more accurate than the target performance used n DDBD. Research on DDBD of brdges (Kowalsky, 2002; Dwar and Kowalsky, 2006; Ortz, 2006; Suarez and Kowalsky, 2008a) has shown good agreement between the target performance used n DDBD and the performance smulated by ITHA. Ths has proven that DDBD s an effectve desgn method and therefore, n general, desgns done wth DDBD do not requre evaluaton by ITHA or other method. Exemptons to ths are the cases were the brdges have characterstcs that cannot be captured by DDBD or have not yet been studed for ther mplementaton n DDBD. Addtonally, ITHA may be requred by the owner for mportant/essental brdges. The condtons under whch ITHA may be requred for desgn evaluaton are: Brdges wth more than sx spans Brdges wth more than two nternal expanson jonts n the superstructure Curves brdges where the subtended angle s greater than 90 o Hghly rregular confguratons When the target dsplacement and target ductlty of crtcal components cannot be accurately assessed. Ths may happen for some substructures that are not consdered n Table 3 or for drlled shaft bents when sol condtons are dfferent of those assumed for the defnton of the equvalent model The program ITHA-Brdge has been developed to perform ITHA of hghway brdges. Ths program s a pre-processor and post-processor of OpenSees (Mazon et al, 2006) and has the followng features: 45

46 From a basc nput automatcally generates the brdge model fles for OpenSees. Supports the substructures shown n Table 3, ntegral and seat-type abutments. Also supports superstructure jonts and plan curvature. Multple acceleraton records can be run n batch mode automatcally. Checks convergence errors n soluton and adjusts the analyss tme step f necessary to acheve convergence. Produces and output fle combnng the output of the dfferent acceleraton records that were run. ITHA-Brdge and ts documentaton can be accessed and used on-lne though the Vrtual Laboratory for Earthquake Engneerng ( 5. APPLICATION EXAMPLES 5.1 Superstructure target dsplacement Fgure 19 Two span contnuous brdges Ths example llustrates the effect of superstructure dsplacement capacty n the ductlty of the central per of the brdge shown n Fg. 19. It s assumed that the dsplacement capacty of the contnuous superstructure s controlled by the yeldng of the longtudnal renforcement n the concrete deck. The yeld dsplacement of the superstructure s computed at the locaton of the central per. The length of the brdge s Ls = 70 m and the wdth of the superstructure deck s w s = 13 m, as shown n Fg