A displacement-based seismic design procedure for concrete bridges ...

Bull Earthquake Eng (2011) 9:537–560 DOI 10.1007/s10518-010-9215-5 ORIGINAL RESEARCH PAPER

A displacement-based seismic design procedure for concrete bridges having deck integral with the piers Vasilios G. Bardakis · Michael N. Fardis

Received: 8 February 2010 / Accepted: 29 September 2010 / Published online: 23 October 2010 © Springer Science+Business Media B.V. 2010

Abstract A multi-step displacement-based seismic design procedure is presented and exemplified for concrete bridges having piers integral with a continuous prestressed deck, which is restrained transversely at the abutments but free to slide there longitudinally. It includes a simple estimation of inelastic deformation demands (chord or plastic hinge rotations in piers, curvatures for the deck) via elastic 5%-damped modal response spectrum analysis and normally entails very few design-analysis iterations. Prescriptive detailing and minimum reinforcement rules are relaxed. Instead, the vertical reinforcement of piers is tailored to seismic deformation demands for uniform ductility ratios across the piers, while their transverse reinforcement is detailed on the basis of explicit checks of their integrity under the design earthquake. The procedure is applied to eight different bridges with three or five spans, box girder deck and piers of various cross-sections with about equal or very different heights. Nonlinear response-history analyses are carried out under ground motions well beyond the design earthquake, in order to evaluate the seismic performance of the resulting designs and compare with that of force-based Eurocode 8 designs of the same bridge. The displacementbased procedure is shown to give more cost-effective and balanced designs than Eurocode 8, without loss in seismic performance. Keywords Concrete bridges · Concrete piers · Displacement-based design · Eurocode 8 · Seismic design · Seismic design of bridges

1 Introduction Recent years have seen the emergence of Displacement-based design (DBD) as the most promising basis for future developments in seismic design. In DBD seismic displacements and deformations are the primary response variables for the design and capacity-demand comparisons for ductile members are expressed in terms of them, instead of forces.

V. G. Bardakis · M. N. Fardis (B) University of Patras, P.O. Box 1424, 26504 Patras, Greece e-mail: [email protected]

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Bull Earthquake Eng (2011) 9:537–560

DBD concepts have advanced more for buildings than for bridges. Appendix I-Part 2 (“Tentative Guidelines for PBSE: Part B Force-displacement approach) in SEAOC (1999) summarizes the State-of-the-Art of DBD for codified design of new buildings, in terms of a “Direct” DBD procedure and an “Equal-displacement” one. The former, proposed by (Priestley 1995; Priestley and Calvi 1997; Priestley et al. 2007; Kowalsky et al. 1995), etc., starts with the conversion of code-specified interstorey drift limits that depend on material, structural system and seismic hazard (i.e. design earthquake level), to a global target displacement, Δt , of an “equivalent” SDOF system. Conversion factors are given for regular buildings, depending on the type of the structural system and the number of storeys. The displacement spectrum of the design earthquake is then entered at this displacement level for a code-specified damping value, which again depends on material, structural system and hazard level. An effective period at peak displacement is read from the spectrum and used (for given total mass) to calculate the secant global lateral stiffness to the peak displacement. The required base shear strength is determined then as the product of this secant stiffness and the target global displacement. The base shear is distributed over the system and its members are dimensioned for the resulting strength demands as in conventional force-based design (FBD). Pushover analysis is used in the end to check that interstorey drift limits or ductility supplies of deformation-controlled members are not exceeded. The procedure concludes with capacity-design dimensioning of brittle members or failure modes, to ensure that the desired yield mechanism is achieved. The “Equal-displacement” DBD procedure in the “Tentative Guidelines” of SEAOC (1999) has the same initial step as the “Direct” one (i.e., calculation of the target global displacement Δt ) and the same concluding stages after the design base shear is determined. At intermediate stages the “equal displacement” approximation and a 5%-damped elastic displacement spectrum are employed to estimate the initial period (secant-to-yield-point), which is used next to find the elastic global stiffness, K in . The design base shear is calculated then as the ratio of the elastic force, K in Δt , to a code-specified global ductility factor, which depends on material, structural system and hazard level. Member dimensioning avoids overstrength in deformation-controlled elements. It is also checked that in the final design the elastic global stiffness at yielding agrees well with the originally determined value, K in . For displacement-based seismic design of bridges (Kowalsky et al. 1995; Calvi and Kingsley 1995; Kowalsky 2002; Dwairi and Kowalsky 2006; Priestley et al. 2007) have proposed a DBD procedure that differs from the “Direct” one for buildings in the determination of the “equivalent” SDOF system. This system was originally based on a single-mode estimate of the displaced shape of the bridge, an approach that works only for regular bridges. For irregular ones, an extension of the Shibata and Sozen (1976) substitute-structure approach was proposed, with the displaced shape based on iterative modal analyses with pier stiffness reduced by the pier ductility factor. These proposals were even extended to isolated bridges, from where the idea of iterative design using a secant-to-peak stiffness and a damping value consistent with peak displacement derives. The Caltrans (2006) Seismic Design Criteria, which embraced DBD as the basis for the design of new bridges, opted for the “Equal Displacement” version of DBD in SEAOC (1999). In the Caltrans (2006) approach the global displacement demand Δt is determined from a 5%-damped elastic spectrum on the basis of the initial period (secant-to-yield-point). Pushover analysis is used then to check that plastic hinge rotation capacity is not exceeded anywhere in the system. Member displacement ductility ratio demands when the target displacement Δt is attained may not exceed values from 4 to 5. Design is essentially a sequence of trial-and-error evaluations via pushover analysis.

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The present paper elaborates and applies a multistep DBD procedure for concrete bridges having yielding piers (i.e. not with massive wall-type piers) integral with a continuous prestressed deck, which is restrained transversely at the abutments but free to slide there longitudinally without significant effects of any impact with the abutment back-wall (Bardakis 2007). The procedure is well integrated with the usual design practice of such bridges for gravity loads and entails the simplest possible estimation of pier inelastic deformation demands, namely through 5%-damped modal response spectrum analysis and the “equal displacement” approximation for member deformations, without recourse to pushover analysis to determine the correspondence between the global displacement demand Δt and local deformation demands. In both these respects it is much closer to the “Equal Displacement” version of DBD than to the “Direct” one, for which much work has already been done by Priestley et al. as outlined above. The procedure is applied for evaluation purposes to eight bridges having a prestressed box girder deck over three or five spans integral with piers of similar or of very different heights. The present proposal bears certain similarities with that of (Panagiotakos and Fardis 1999, 2001) for DBD of buildings. The most important one is that in both of them detailing of plastic hinges is carried out to meet the deformation demands due to the design earthquake (the 475 year one for structures of ordinary importance). However, there are important differences as well, not only due to the fundamental differences between buildings and bridges and the associated seismic design requirements and procedures. A major one is that the present procedure (in its Step 4) tailors the vertical reinforcement of the piers to the seismic deformation demands of the design earthquake itself for uniform ductility ratios across all piers; by contrast, member longitudinal reinforcement in the buildings of (Panagiotakos and Fardis 1999, 2001) is dimensioned for the force demands of a serviceability earthquake, several times smaller than the design earthquake.

2 Proposed displacement-based seismic design procedure The proposed procedure consists of several steps: Step 1: Dimensioning of the deck and the piers for: • the Ultimate Limit State under the combinations of factored permanent and traffic loads at all relevant stages of construction as well as in the completed bridge, taking into account the redistribution of action effects due to creep and losses of prestress, etc., as appropriate; • the Serviceability Limit State under the pertinent combinations of permanent and transient loadings in the completed bridge. Step 2: Estimation of the effective stiffness of the deck and the piers, (E I )eff , which is representative of the elastic stiffness for the seismic response analysis under the design earthquake in Step 3. For this analysis the deck and each pier are discretized longitudinally as a string of prismatic beam elements in 3D. • For the deck, for which the almost final reinforcement and prestressing has been dimensioned in Step 1: – For bending about the horizontal axis (about which the section and its prestressing are normally asymmetric), (E I )eff is based on a bilinear idealization of the monotonic M-ϕ diagram with different post-elastic slopes in positive or negative bending. So, it is taken as the slope of the line connecting the points of (a) decompression

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/cracking of the extreme fibres on the same side of the section as the mean tendon and (b) yielding of the tension reinforcement on the side opposite to the mean tendon. – For bending about the vertical axis (an axis of symmetry), (E I )eff may be taken equal to the first slope of a bilinear fitting of the M-ϕ diagram. It comes out lower (e.g., by 10 to 20 %) than the elastic stiffness of the uncracked gross concrete section. For an element in the discretization of the deck, the average stiffness of its two end sections is used. • For the piers, (E I )eff is the secant stiffness to the yield point of the full pier, defined as: E Ie f f =

My L s 3θ y

(1)

where My and L s denote the yield moment and the moment-to-shear-ratio (“shear span”) at the yielding end of the pier and θy the chord rotation there at yielding, i.e., the deflection of the end of the shear span divided by L s , given by (Biskinis and Fardis 2010a): – For solid rectangular piers with depth h: L s + aV z h θy = ϕy (2a) + 0.0014 1 + 1.5 + θ y,sli p 3 Ls – For wall-type or hollow rectangular piers: θy = ϕy

L s + aV z + 0.0013 + θ y,sli p 3

(2b)

– For circular piers of diameter D: θy = ϕy

Ls L s + aV z + 0.0022 max 0; 1 − + θ y,sli p 3 6D

(2c)

In Eq. (2) ϕy is the yield curvature of the end section (from plane section analysis with elastic σ − ε relations till yielding of the tension or the compression chord), av z is the tension shift due to 45◦ cracking, with z the internal lever arm and av a zero-one variable: av = 0 if the shear force at diagonal cracking, VRc (i.e., the shear resistance of members without shear reinforcement in Eurocode 2), exceeds the yield force at flexural yielding (i.e. if VRc > My /L s ); av = 1 otherwise. The fixed-end rotation of the end section, due to slip of tension bars (of mean diameter dbL ) from their anchorage beyond the member end in the foundation or the joint, is (Biskinis and Fardis 2010a): θ y,sli p =

ϕ y dbL f y L with f yL and f c in MPa √ 8 fc

(3)

Yielding is assumed to take place at the end sections where plastic hinging is expected under the design earthquake and the concurrent gravity loads. Single piers work in the transverse direction of the bridge as vertical cantilevers, with plastic hinging only at the base; their full clear height may be taken as the shear span, L s . In all other cases, including bending of a pier in the longitudinal direction of the bridge, plastic hinging will normally take place at the base of the pier and at its connection to the deck; L s may then be taken as half the pier full clear height. To apply Eqs. (1) and (2) one needs to know the geometry of the pier section, the pier axial load N and the value of L s , but also the amount and layout of the vertical reinforcement in the section. The latter is normally not available during Step 2 of the design procedure, as the piers have not been dimensioned yet for earthquake. In that case a preliminary estimate of (E I )eff may

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obtained from an empirical expression in (Biskinis and Fardis 2010a) in terms only of N , L s and geometric parameters of the pier section: (E I )eff Ls N = a 0.8 + ln 1 + 0.048 (4) (MPa) E c Ic h Ac where Ic , Ac and h are the moment of inertia, the area and the depth of the pier section, E c the Elastic Modulus of concrete and coefficient a is equal to 0.081 for circular or rectangular piers and to a = 0.09 for hollow rectangular ones. Note that, unlike My and θy in Eq. (1), which are calculated from first principles, possibly with calibration to test results, Eq. (4) strictly applies within the range of parameter values it has been statistically fitted to in (Biskinis and Fardis 2010a): – in circular piers: shear span ratio, L s /D, from 1.0 and 8.5 (mean value 3.25), concrete strength, f c , between 19 and 90 MPa (mean value about 35 MPa), axial load ratio, N /Ac f c , between −0.1 and 0.7 (mean value about 0.15) and vertical reinforcement ratio between 0.5 and 5.7% (average: 2.5%); – in hollow rectangular piers: shear span ratio, L s / h, between 0.6 and 8.3 (mean value 2.6), N /Ac f c between 0 and 0.5 (average about 0.075), wall thickness, bw , from 50 to 500 mm (average just 120 mm), wall slenderness, h/bw , from 2.5 to 36 (mean value 12.5), vertical reinforcement ratio between 0.34 and 6.2% (average 1.35%) and f c between 20 and 102 MPa (mean value 43 MPa). The calculation of stiffness for the deck and the piers should be based on mean values of material properties. Step 3: Estimation of (inelastic) deformation demands (chord rotations, θEd , at pier ends, deck curvatures, ϕEd ) for the two horizontal components of the design earthquake from the 5%-damped elastic spectrum. It is essential to account in this calculation for the dynamic characteristics of the bridge. At a minimum, modal response spectrum analysis with Complete Quadratic Combination (CQC) of modal responses should be used. If the computational capability for response-history analysis is available (e.g., for evaluation of the final design through nonlinear dynamic analysis), elastic response-history analysis with at least seven spectrum-compatible motions per horizontal direction may be used instead. For bridges with piers of about equal height, modal response spectrum analysis with 5% damping provides on average a fair and unbiased estimate of inelastic deformation demands. In the companion paper (Bardakis and Fardis 2010) six such bridges (C3, C3-a, T6, T6-a, T6-b and T6-c, considered in Sect. 3 below) designed to Eurocode 8 for PGA on rock of 0.14 g, have been analysed for suites of ground motions with PGA on rock of 0.125 g, 0.25 g, 0.35 g or 0.45 g; the average ratio of inelastic deformations from nonlinear dynamic to those from a 5%-damped modal response spectrum analysis decreases as follows with increasing PGA: • for longitudinal earthquake: – from 1.04 to 0.90 at pier ends (average: 0.97), – from 1.13 to 0.87 in the deck (average: 1.03); • for transverse earthquake: – from 1.06 to 0.90 at pier ends (average: 0.98); – from 1.29 to 0.98 in the deck (average: 1.11). The average ratios slightly drop if the six bridges are designed according to the present DBD procedure: • for longitudinal earthquake:

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– from 1.00 to 0.90 at pier ends (average: 0.94); – from 1.12 to 0.79 in the deck (average: 0.95); • for transverse earthquake: – from 1.07 to 0.90 at pier ends (average: 0.98); – from 1.28 to 0.97 in the deck (average: 1.10). By contrast, for bridges C3-b, T6-d in Sect. 3 below, with very different piers heights, 5%-damped modal response spectrum analysis underestimates the inelastic deformations obtained from a nonlinear dynamic one: for design to Eurocode 8 for PGA on rock of 0.14 g, the average ratio quoted above changes as follows, when the PGA on rock increases from 0.125 g to 0.45 g (intermediate values 0.25 g, 0.35 g): • for longitudinal earthquake: – from 1.12 to 1.04 at pier ends (average: 1.10), – from 1.25 to 1.20 in the deck (average: 1.26); • for transverse earthquake: – from 1.34 to 1.12 at pier ends (average: 1.22); – from 1.35 to 1.00 in the deck (average: 1.15). The average ratios slightly drop if these two bridges are designed according to the present DBD procedure: • for longitudinal earthquake: – around 1.08 at pier ends; – from 1.32 to 1.06 in the deck (average: 1.24); • for transverse earthquake: – from 1.24 to 1.11 at pier ends (average: 1.18); – from 1.32 to 1.00 in the deck (average: 1.14). Step 4: Selection of a target value of the chord rotation ductility factor, μθ , (defined as chord rotation demand divided by chord rotation at yielding of the end section) at those pier ends where plastic hinges are expected to develop under the design earthquake. Then the target chord rotation at yielding, θy , is estimated for each direction of bending of the pier from the value obtained in Step 3 for the chord rotation demand, θEd , in that direction: θy = θEd /μθ

(5)

In this way a uniform level of inelasticity may be targeted in piers of different heights. Then, the pier yield moment, My , corresponding to the value of θy from Eq. (1) is estimated by inverting Eq. (1) for My , using there the same values of (E I )eff and L s as in Step 2. The pier vertical reinforcement is then dimensioned for this value of My . The objective of this step is to achieve as uniform a distribution of inelasticity in the piers as possible and to avoid overstrengths. If it turns out that the vertical reinforcement at one or more likely plastic hinge locations in the piers is governed by Step 1 in lieu of the present step, it is recommended to reduce the target value of μθ in the corresponding direction of bending until Step 4 finally governs the vertical reinforcement at all likely plastic hinges in the piers. It is worth noting that for shorter piers, which in general have larger values of θEd , Eq. (5) will also give larger θy values and—if Eq. (4) is used for (E I )eff —proportionally larger values of My and hence more pier reinforcement. The effect of the larger reinforcement ratio on the (E I )eff -value of the shorter piers will be taken into account in the iterations to be carried out during Step 9, below.

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Step 5: Ensuring that the deck will not yield under the combination of the curvatures due to the design earthquake, ϕEd , and those induced by the concurrent quasi-permanent actions, ϕG+P : ϕG+P ± γo ϕEd ≤ ϕy

(6)

In Eq. (6) ϕy is the deck yield curvature from the M-ϕ diagram and γo a pier overstrength factor. Values of γo between 1.25 and 1.35 take into account sufficiently strain-hardening in the pier. Equation (6) is checked separately in the two directions of bending of the deck under the corresponding horizontal component of the design earthquake. It expresses in terms of deformations, in lieu of moments, the verification that the deck remains elastic under the design seismic action. To satisfy Eq. (6) at all sections of the deck it may be necessary to increase the prestress and/or the reinforcement over those required for Step 1. Step 6: Verification of the piers against the chord rotation demands from Step 3 for both horizontal components of the design seismic action: θEd ≤ θRd = θuk,0.05 /γRd

(7)

In Eq. (7) θRd is the design value of the pier chord rotation capacity, θuk,0.05 , is the 5%-fractile of the pier ultimate chord rotation and γRd a safety factor against exceedance of the ultimate chord rotation. The expected value of the pier ultimate chord rotation, θum , of circular piers may be obtained from (Biskinis and Fardis 2006); that of rectangular (including wall-like) or hollow rectangular piers from,(Biskinis and Fardis 2010b): h θu,m = θ y + 0.017 1 − 0.05 max 1.5; min 10; (0.2)ν bw αρw f yw max(0.01; ω2 ) L s 1/3 0.2 fc f c 25 (8) max(0.01; ω1 ) h where: – θy from Eqs. (2a), (2b); – bw : width of one web, even in cross-sections with two or more parallel webs; – ν = N /bh f c , with b: width of compression zone, N : axial force (positive for compression); – ω1 = (ρ1 f y1 + ρv f yv )/ f c : mechanical reinforcement ratio of the entire tension zone, including the tension chord (index 1) and the web longitudinal bars (index v); – ω2 = ρ2 f y2 / f c : mechanical reinforcement ratio of the compression zone; – L s / h = M/V h: shear-span-to-depth ratio at the section of maximum moment; – ρw = Ash /bw sh : ratio of transverse steel parallel to the plane of bending; – α: confinement effectiveness factor: 2

bi /6 sh sh 1− α = 1− (9) 1− 2bo 2h o bo h o with sh denoting the centreline spacing of stirrups, bo and h o the confined core dimensions to the centreline of the hoop and bi the centreline spacing along the section perimeter of longitudinal bars (indexed by i) engaged by a stirrup corner or a cross-tie. The scatter of the data about θum is such that the 5%-fractile of the pier ultimate chord rotation is: θuk,0.05 = 0.5θum .

(10)

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If the design seismic action has 10% probability of being exceeded in 50 years, an appropriate value is γRd = 2. The pier confining reinforcement is the main free dimensioning variable governing θum and should be chosen so that Eq. (7) is met. Step 7: Verification in shear of the piers and the deck-to-pier connections, increasing their size - if needed - and dimensioning their shear reinforcement. • The requirements for the piers are: – they safely sustain the degradation of shear resistance in their plastic hinge(s) due to the cyclic inelastic deformations predicted there from the analysis in Step 3 (which are expressed for the present purposes by the chord rotation ductility factor at the corresponding pier end, μθ ): VCD,p ≤ VRd,cyc (μθ )/γRd

(11)

– brittle shear failure before plastic hinging is prevented; to this end each pier is dimensioned so that it satisfies all along its height: VCD,p ≤ VRd,mon /γRd

(12)

Eqs. (11), (12) are checked separately in each one of the two orthogonal transverse directions of the pier, with: – VCD,p : “capacity design” shear force of the pier, determined from equilibrium, assuming that the pier develops its overstrength moment capacity, γo MRd , only at the base for single piers bent in the transverse direction or at both top and bottom sections in all other cases; MRd is the design value of the pier moment resistance and γo the overstrength factor introduced in Step 5 above. – VRd,cyc (μθ ): design value of cyclic shear resistance after flexural yielding, taken to decrease with increasing chord rotation ductility factor of the pier, μθ , e.g., as in (Biskinis et al. 2004): ◦ For diagonal tension: h−x V Rs,d,cyc μθ = min (N ; 0.55Ac f cd ) + 1 − 0.05 min 5; μθ − 1 . 2L s

Ls 0.16 max(0.5; 100ρtot ) 1 − 0.16 min 5; (13) f cd Ac + Vwd h ◦ for diagonal compression in the web of hollow rectangular or of wall-like piers: N 1 + 1.8 min(0.15; V R max,d,cyc μθ = 0.85 1 − 0.06 min 5; μθ − 1 Ac f cd Ls min(100 MPa; f cd )zbw · (1 + 0.25 max(1.75; 100ρtot )) 1 − 0.2 min(2; h

(14) where units are MN and m and: – the axial force N is taken positive for compression and zero for tension, – x is the neutral axis depth, – ρtot is the total longitudinal reinforcement ratio, – L s / h is the shear-span-to-depth ratio, – z is the internal lever arm of a rectangular section, – Ac = bw d for sections with rectangular webs of total width bw and effective depth d, and Ac = Dc2 /4 in circular ones with diameter of concrete core inside the hoops Dc

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– Vwd is the contribution of transverse steel to the resistance in diagonal tension in a 45◦ -truss analogy: ◦ for rectangular webs with transverse steel ratio ρw : Vwd = ρs bw z f ywd

(15a)

◦ for circular sections with diameter D, and circular hoops with cross-sectional area, spacing and concrete cover Asw , sh and c, respectively: Vwd =

π Asw f ywd (D − 2c) 2 sh

(15b)

Once the vertical reinforcement of the pier has been dimensioned in Step 4, μθ may be taken equal to the value of θEd at the end section of the pier from Step 3, divided by the chord rotation at yielding there, θy , as this finally comes out of Step 4 and of the dimensioning of the pier’s vertical reinforcement there. As noted in Steps 2 and 9, the value of θy for the pier depends on the amount and layout of its vertical reinforcement, as it comes out of Step 4, and on the shear span L s (: moment-to-shear ratio), as this comes out of the elastic analysis in Step 3, see, Eq. (2); – VRd,mon : design value of the shear resistance for monotonic loading (as for non-seismic actions); it may be taken equal to VRd,cyc (μθ = 1), with VRd,cyc (μθ ) defined as above; – γRd is a safety factor against exceedance of the shear force resistance, for which a value γRd = 1.25 seems appropriate. Being design values of the shear force resistance, the values of VRd in Eqs. (11)–(15) are computed using design strengths for the materials (nominal strengths divided by the material partial factors) or applying an overall capacity reduction factor on the nominal shear resistance. • A capacity design vertical shear force, VCD,jv , is also computed for each deck-to-pier connection, assuming that, at the face of the connection, the pier and its tension chord develop their overstrength capacities, γo MRd , and γo As,1 f yd , respectively, where As,1 is the cross section of the reinforcement in the tension chord of the pier and f yd its design yield stress: VCD,jv = γo As,1 f yd − Vmin,deck

(16)

In Eq. (16) Vmin,deck is the algebraically minimum shear force that develops in the deck on either side of the connection when a plastic hinge develops at the top of the pier: Vmin,deck ≈

γo M Rd H − max Vg,l ; Vg,r L Hcl

(17)

In Eq. (17) H and Hcl are the theoretical and the clear pier height respectively, L the average pier-to-pier distance on either side of the connection and Vg,l the shear force in the deck on the left (index: l) or on the right (index: r ) face of the connection, due to the gravity loads concurrent with the seismic action. The core of the connection is then verified and its shear reinforcement dimensioned, on the basis of the shear stress and the horizontal and vertical normal stresses at its centre, e.g., according to Eurocode 8 (CEN 2005). Step 8: Verification of the deck in shear and torsion. To meet this verification, it may be necessary to increase the deck cross-section and/or dimension its transverse reinforcement

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for the shear(s) and torque due to the quasi-permanent actions concurrent with the design earthquake (collectively denoted here as E G+P ) combined with the overstrength shear(s) and torque due to the design seismic action. These latter seismic action effects are obtained by dividing their elastic estimates from Step 4 (collectively denoted here as E Ed ) by the target pier chord rotation ductility factor, μθ , used in Step 4: E d = E G+P ± γo E Ed /μθ

(18)

Step 9: Updating the secant stiffness to the yield point of the pier, (E I )eff , using Eq. (1). The values of My , θy , L s to be used there should be consistent with the amount and layout of vertical reinforcement from Step 4 and the moment-to-shear ratio at the end of the pier in the plane of bending considered, from the elastic analysis of Step 3 for the horizontal earthquake component closest to the plane of bending. (If Step 3 uses elastic response-history analysis, the ratio of the average moment to the average shear is used, from the analyses for the different motions in the horizontal direction considered). If Eq. (1) is applied considering plastic hinging at both ends of a pier and the vertical reinforcement or the shear span differs between these two ends, the average outcome of Eq. (1) for the two ends is used for the entire pier. If the so-computed (E I )eff -values of some piers deviate significantly from those estimated in Step 2 and used for the calculation of seismic deformation demands in Step 3, then Steps 3 to 8 are repeated until the values of (E I )eff from Step 9 agree (within a certain tolerance) with the effective stiffness values used in the analysis. Step 10: Evaluation of the design through nonlinear dynamic analysis and revision wherever needed. For bridges with piers of about equal height this step is optional. However, it is essential for bridges having piers with very different heights, for which the linear elastic analysis with 5% damping often underestimates inelastic deformation demands (Bardakis and Fardis 2010). Step 4 seems equivalent to the reduction of the pier internal forces from the analysis with a 5%-damped elastic spectrum in force-based design, with μθ playing the role of the force reduction factor R or the behaviour factor q of Eurocode 8. However, it differs from it in two important aspects: • Step 4 does not aim at providing a margin between elastic and inelastic force demands which is at least equal to the reduction factor (μθ , R or q) but in reality is larger and nonuniform owing to detailing and minimum requirements or to design of the pier vertical reinforcement for non-seismic actions. The goal of Step 4 is, instead, to provide all piers with a uniform margin as close as possible to μθ . • Although Eq. (5) may seem to express in terms of deformations the division of elastic seismic forces by R or q to obtain the yield force, numerically μθ is not equal to R or q: the effective stiffness values from Steps 2 and 9 are much smaller than those typically used in conventional, force-based design. So, they result in longer natural periods and correspondingly lower elastic spectral accelerations. Therefore, Step 4 essentially provides a reduction factor on conventional design forces about equal to μθ times the ratio of the elastic spectral acceleration resulting from the stiffness of conventional, force-based design to the spectral acceleration corresponding to the effective stiffness from Steps 2 and 9. It is worth noting that Steps 4 and 6 essentially provide all pier ends with a chord rotation ductility factor, which is equal to γRd μθ if it refers to the 5%-fractile of the pier ultimate chord rotation, θuk,0.05 , or to 2 γRd μθ if it refers to its mean (expected) value, θum .

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3 Application of proposed procedure and evaluation of the designs The DBD procedure outlined above has been applied for the seismic design of eight bridges with integral deck and piers, described in the companion paper (Bardakis and Fardis 2010) and depicted in Figs. 2 and 3 therein. The design seismic action is synchronous under all piers or abutments, with peak ground acceleration (PGA) equal to 0.14 g on rock times 1.15 on top of firm soil, giving a design PGA of 0.16 g over firm soil. Besides, an importance factor of 1.3 is applied to the ground motion of the three balanced cantilever, type-C3, bridges. To emphasize the difference in the outcome of the design from a conventional, force-based one, e.g., according to Eurocode 8 (CEN 2005), the design of its piers has been taken to the extreme: their minimum vertical reinforcement has been considered equal to just 0.2%, as given in Eurocode 2 for design against non-seismic actions. Table 1 compares the amount of pier reinforcement of the DBD bridges (the vertical one sometimes being controlled by Step 1 of the design) to that from conventional, force-based design (FBD) of the same bridge according to Eurocode 8. For the very little vertical reinforcement chosen here for the DBD piers, their secant stiffness to the yield point is overestimated by the reinforcement-independent empirical expression of Eq. (4), which was fitted to test data from specimens with much higher steel ratios. So, Steps 3 to 8 were repeated after updating the values of (E I )eff in Step 9 on the basis of the more accurate Eq. (2) and the actual pier reinforcement. One design-analysis iteration was normally sufficient; a second iteration was necessary for few of the bridge designs. The iterations stopped when there was no substantial change in the amount of reinforcement from one iteration to the next. Table 2 shows that the final value of the fundamental period of the DBD version of each bridge, Tm , is longer than that from Step 2, To ; Tm is also longer in the DBD than in the FBD version of each bridge. Global displacement demands are accordingly larger for the same seismic action. For the FBD bridges Table 2 lists also the conventional periods according to Eurocode 8, Td , computed using for the deck the uncracked gross section stiffness and for the piers the ratio of moment-to-curvature at yield point (neglecting the effects of shear deformation and inclined cracking, or of bond slip of vertical bars from their anchorage beyond the end section). The period values suggest that in the FBD bridge versions the effective stiffness from Eq. (4) is on average 60% of that obtained according to Eurocode 8; for the lighter reinforced DBD ones, it is about 40% of the Eurocode 8 values. The performance of the two bridge designs has been compared on the basis of nonlinear response-history analyses, with input seven motions emulating historic earthquakes as described in the companion paper (Bardakis and Fardis 2010). The motions are scaled to a PGA on rock of 0.25 g, 0.35 g or 0.45 g (times 1.15 on soil top and, for the type-C3 bridges, times the importance factor of 1.3). So, the analyses take place under a PGA from almost twice to over three-times the design earthquake. The expected values of material strengths used in the analysis are given in (Bardakis and Fardis 2010). The analysis program used, ANSRuop-Bridges, and the modelling are also outlined there. The seismic performance of the bridges is evaluated and compared in Figs. 1, 2, 3, 4, 5, and 6 and Tables 3, 4, 5, 6, 7, and 8 on the basis of the following three measures of response and performance (mean value over the seven input motions, each applied once in the transverse direction, or with plus or minus in the longitudinal): • The peak value of the deformation ductility factor, μθ : for the chord rotation at pier ends or for the curvature along the deck (Figs. 1, 2; Tables 3, 4).

123

123

T6-a

T6

C3-b

C3 and C3-a

Bridge

1.05

0.67

0.88

0.34

0.44

1.61

1.57

1.58

0.88

0.89

1.56

0.55

0.55

1.00

1.57

ρw,T

ρw,L

ρw,T

ρL

ρw,L

ρw,T

ρw,L

ρw,T

ρL

ρw

ρw

ρL

ρw

Top half

Bottom half

Plastic hinge

Outside plastic hinge

In plastic hinge


In plastic hinge

ρw,L

Box or solid piers

Location

ρL

Force-based

Design

–

–

–

–

–

0.23

0.24

0.46

0.48

1.00

–

–

–

–

–

Twin blades (30 m)

0.81

0.51

0.37

0.29

0.24

0.89

0.88

1.58

1.57

1.61

0.46

0.37

0.51

0.41

0.76

0.43

0.32

0.21

0.36

0.29

0.44

0.36

0.48

–

–

–

–

–

0.23

0.24

0.46

0.48

0.68

–

–

–

–

–

–

–

–

–

–

0.20

0.21

0.40

0.42

0.39

–

–

–

–

–

Bottom half

Top half

Longer

Shorter

Outer long piers–twin blades (30 m)

Box or solid piers

Displacement-based

–

–

–

–

–

0.23

0.24

0.46

0.48

0.68

–

–

–

–

–

Top half

–

–

–

–

–

0.20

0.21

0.40

0.42

0.39

–

–

–

–

–

Bottom half

Central long piers–twin blades (30 m)

–

–

–

–

–

0.23

0.24

0.46

0.48

0.68

–

–

–

–

–

Top half

–

–

–

–

–

0.20

0.21

0.40

0.42

0.39

–

–

–

–

–

Bottom half

Short piers–twin blades (30 m)

Table 1 Pier reinforcement ratios (%): vertical, ρL , horizontal and confining: ρw,L in longitudinal direction of the bridge ρw,T in transverse direction (denoted by ρw if ρw,L = ρw,T )

548 Bull Earthquake Eng (2011) 9:537–560

T6-d

T6-c

T6-b

Bridge


In plastic hinges


In plastic hinges

Plastic hinge Outside plastic hinge

0.70 0.68

0.40

0.20

1.00

0.57

0.57

0.51

0.51

1.00

0.37

0.35

0.19

0.18

ρw

ρw

ρL

ρw,L

ρw,T

ρw,L

ρw,T

ρL

ρw,L

ρw,T

ρw,L

ρw,T 0.33

0.34

1.05

–

–

–

–

–

–

–

–

1.00

ρL

–

0.79

Twin blades (30 m)

ρw

Box or solid piers

Location


Force-based

Design

Table 1 continued

0.18

0.19

0.35

0.37

1.00

0.32

0.26

0.36

0.29

0.51

0.14

0.27

0.21

0.41

0.24

0.25

0.48

0.50

0.88

–

–

–

–

–

–

–

–

–

0.13

0.14

0.26

0.27

0.26

–

–

–

–

–

–

–

–

–

Bottom half

Top half

Longer

Shorter

Outer long piers–twin blades (30 m)

Box or solid piers

Displacement-based

0.17

0.18

0.33

0.35

0.54

–

–

–

–

–

–

–

–

–

Top half

0.14

0.15

0.28

0.30

0.21

–

–

–

–

–

–

–

–

–

Bottom half

Central long piers–twin blades (30 m)

0.17

0.18

0.33

0.35

0.41

–

–

–

–

–

–

–

–

–

Top half

0.14

0.13

0.28

0.27

0.26

–

–

–

–

–

–

–

–

–

Bottom half

Short piers–twin blades (30 m)

Bull Earthquake Eng (2011) 9:537–560 549

123

123

1.47

1.82

2.85

1.16

1.21

2.61

T6-a

T6-b

T6-c

T6-d

4.64

72.2

92.3

97.5

97.2

88.6

80.8

91.6

92

2.05

1.75

1.48

2.02

1.02

0.70

3.11

2.38

2.04

1.91

2.17

1.05

0.86

4.08

3.72

2.72

73.7

62

75.4

81.1

80.6

82.9

59

72.6

2.25

3.03

1.41

1.26

2.59

0.59

4.51

2.18

4.28

2.06

1.86

3.75

0.93

5.36

2.84

2.97

Tm (s)

71.4

92.8

97.6

97.2

98.4

80.7

94

94.1

Particip. mass (%)

2.41

1.55

1.83

1.10

0.83

3.77

3.37

3.03

To (s)

Transverse

3.17

2.04

2.41

1.13

1.11

4.38

4.03

3.75

Tm (s)

56.6

75.5

81

81

82.4

62.6

74.7

76.5

Particip. mass (%)

(Td : periods in FBD from the conventional stiffness values in Eurocode 8; To : initial estimate of period in Step 2 of DBD using the empirical stiffness values for piers; Tm : final estimate of period based on the pier and deck reinforcement)

3.26

3.33

0.62

3.94

0.41

C3-b

T6

2.65

To (s)

2.80

1.89

1.84

C3

C3-a

Particip. mass (%)

Td (s)

Tm (s)

Td (s)

Tm (s)

Longitudinal

Transverse

Longitudinal

Particip. mass (%)

Displacement-based design (DBD)

Force-based design (FBD)

Table 2 Fundamental periods of the study bridges in FBD and DBD version

550 Bull Earthquake Eng (2011) 9:537–560


551

Fig. 1 Chord-rotation ductility ratios at pier ends from nonlinear dynamic analysis (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs

Fig. 2 Peak deformation demand from nonlinear dynamic analyses divided by value at departure from linearity (rebar yielding of the deck opposite to the mean tendon, or deck decompression on the side of the mean tendon) along the deck (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs

Fig. 3 Chord-rotation demand-to-supply (damage) ratios at pier ends from nonlinear dynamic analysis (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs

123

552


Fig. 4 Chord-rotation demand-to-supply (damage) ratios along the deck from nonlinear dynamic analysis (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs

Fig. 5 Shear force demand-to-supply (damage) ratios at pier ends from nonlinear dynamic analysis (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs Table 3 Average over all pier ends of peak chord rotation demand divided by chord rotation at rebar yielding PGA = 0.25g

PGA = 0.35g

PGA = 0.45g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.79

0.88

0.55

0.60

1.05

1.18

0.74

0.81

1.24

1.49

0.94

1.00

C3-a

0.88

0.97

0.62

0.70

1.18

1.25

0.84

0.92

1.50

1.54

1.03

1.12

C3-b

0.85

1.46

0.50

0.57

1.26

2.30

0.64

0.75

2.86

3.29

0.79

0.97

T6

1.16

2.02

1.04

1.52

1.62

2.89

1.08

2.15

2.18

3.28

1.61

2.45

T6-a

0.90

0.66

0.24

0.20

1.17

0.89

0.33

0.27

1.36

1.11

0.43

0.35

T6-b

1.31

1.72

0.87

0.87

1.67

2.20

1.13

1.15

2.01

2.69

1.34

1.41

T6-c

0.76

1.01

0.71

0.80

1.03

1.32

0.90

1.03

1.25

1.61

1.13

1.21

T6-d

0.68

0.99

0.62

0.77

0.98

1.65

0.80

1.00

1.19

2.13

0.95

1.19

Average 0.92

1.21

0.64

0.75

1.24

1.71

0.81

1.01

1.70

2.14

1.03

1.21

123


553

Table 4 Average over the deck of peak deformation demand divided by deformation at departure from linearity (due to rebar yielding at the side of the deck opposite to the mean tendon under longitudinal earthquake, or due to deck decompression in all other cases) PGA = 0.25g

PGA = 0.35g

PGA = 0.45g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.75

0.79

0.56

0.57

1.04

1.06

0.75

0.75

1.17

1.24

0.93

0.93

C3-a

0.45

0.40

0.48

0.53

0.54

0.49

0.64

0.68

0.59

0.54

0.80

0.80 0.89

C3-b

0.34

0.36

0.62

0.60

0.45

0.43

0.76

0.75

0.52

0.47

0.90

T6

0.43

0.19

0.77

0.97

0.48

0.20

0.81

1.36

0.51

0.20

1.21

1.57

T6-a

0.15

0.14

1.43

1.44

0.18

0.19

1.89

1.89

0.19

0.22

2.42

2.35

T6-b

0.26

0.15

0.86

0.83

0.28

0.15

1.10

1.09

0.29

0.16

1.34

1.33

T6-c

0.52

0.44

0.91

0.96

0.76

0.49

1.17

1.21

0.86

0.52

1.40

1.43

T6-d

0.42

0.45

1.61

1.55

0.57

0.56

2.07

2.02

0.66

0.61

2.51

2.39

Average 0.41

0.36

0.90

0.93

0.54

0.45

1.15

1.22

0.60

0.50

1.44

1.46

Table 5 Average over all piers of the flexural damage ratio (ratio of chord rotation demand to ultimate) PGA = 0.25 g

PGA = 0.35 g

PGA = 0.45 g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.15

0.18

0.12

0.14

0.21

0.24

0.17

0.19

0.24

0.30

0.25

0.24

C3-a

0.17

0.22

0.14

0.16

0.23

0.28

0.20

0.24

0.29

0.36

0.24

0.30

C3-b

0.13

0.15

0.09

0.11

0.18

0.21

0.11

0.14

0.22

0.38

0.13

0.17

T6

0.23

0.31

0.20

0.20

0.33

0.43

0.21

0.31

0.45

0.51

0.34

0.36

T6-a

0.12

0.14

0.05

0.06

0.16

0.21

0.07

0.08

0.19

0.27

0.09

0.09

T6-b

0.23

0.25

0.19

0.17

0.29

0.32

0.24

0.22

0.35

0.39

0.29

0.26

T6-c

0.20

0.22

0.19

0.20

0.27

0.30

0.24

0.27

0.33

0.36

0.30

0.32

T6-d

0.09

0.10

0.10

0.14

0.12

0.14

0.13

0.19

0.14

0.17

0.16

0.23

Average 0.17

0.20

0.14

0.15

0.22

0.27

0.17

0.20

0.28

0.34

0.23

0.25

• The damage ratio in flexure, equal to the maximum ratio of flexural deformation demand to the corresponding capacity during the response (Figs. 3, 4; Tables 5, 6). For the deck, the flexural capacity is defined by the point where a tendon yields, or the non-prestressed reinforcement ruptures, or the concrete crushes, whichever happens first in the momentcurvature diagram of the relevant section. The flexural capacity of piers is taken as their expected (mean) ultimate chord rotation, θum , from Step 6 in Sect. 2, computed on the basis of the current values of the pier axial load, N , and of the moment-to-shear ratio (“shear span”) L s . In fact, the same expressions were used in Eq. (7) for the calculation of the confinement reinforcement in Step 6. • The damage ratio in shear, computed as the maximum ratio of shear force demand to capacity during the response (Figs. 5, 6; Tables 7, 8). The shear capacity is the cyclic shear resistance in diagonal tension or compression, VRd,cyc (μθ ), at the current values of

123

554


Table 6 Average over the deck of the flexural damage ratio (: ratio of flexural deformation demand to ultimate) PGA = 0.25 g

PGA = 0.35 g

PGA = 0.45 g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.09

0.09

0.15

0.15

0.14

0.12

0.20

0.20

0.16

0.14

0.24

0.24

C3-a

0.09

0.08

0.14

0.16

0.11

0.10

0.19

0.20

0.12

0.11

0.23

0.23

C3-b

0.07

0.08

0.20

0.19

0.09

0.09

0.24

0.24

0.11

0.10

0.29

0.28

T6

0.08

0.03

0.11

0.13

0.09

0.04

0.11

0.17

0.09

0.04

0.17

0.20

T6-a

0.03

0.02

0.18

0.18

0.04

0.03

0.24

0.24

0.04

0.03

0.30

0.29

T6-b

0.05

0.03

0.12

0.11

0.05

0.03

0.15

0.14

0.05

0.03

0.18

0.17

T6-c

0.09

0.08

0.13

0.13

0.13

0.09

0.16

0.17

0.15

0.09

0.20

0.20

T6-d

0.08

0.09

0.15

0.15

0.11

0.11

0.20

0.19

0.12

0.12

0.24

0.23

Average 0.07

0.06

0.15

0.15

0.09

0.08

0.19

0.19

0.10

0.08

0.23

0.23

Fig. 6 Shear force demand-to-supply (damage) ratios along the deck from nonlinear dynamic analysis (average for 7 motions in transverse, or for 14 in longitudinal direction) in “displacement-based” v force-based designs

N , L s and μθ , from Eqs. (13)–(15), used in Eq. (11) for the verification of the piers in shear during Step 7. The ANSRuop-Bridges keeps track of the damage measures above, using at each point of the response the current values of the axial load, shear span, chord rotation ductility factor, etc., for the calculation of the capacity. The value reported in the end is the most adverse one during the response history. Because: (a) the DBD bridges, being more flexible, are subjected to larger global displacement demands than the FBD designs, and (b) the piers of the DBD bridges have much lower vertical reinforcement ratios than their FBD counterparts and undertake a larger fraction of the greater global displacements, the ductility factors, μθ , are in general larger in the piers of the DBD bridges than in their FBD counterparts. This is especially so for the longitudinal earthquake, for which the relative stiffness and flexural resistance of the deck and the piers affects most the seismic response. By contrast, this earthquake induces much less inelastic action in the deck of a DBD bridge than in its FBD version, although the two decks have absolutely the same reinforcement and prestressing. In other words, the present application

123


555

Table 7 Average over all piers of the shear damage ratio (: ratio of shear force demand to resistance) PGA = 0.25 g

PGA = 0.35 g

PGA = 0.45 g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.43

0.46

0.25

0.27

0.55

0.53

0.28

0.34

0.56

0.56

0.31

0.42

C3-a

0.48

0.57

0.22

0.38

0.55

0.61

0.26

0.40

0.58

0.64

0.33

0.46

C3-b

0.17

0.15

0.18

0.19

0.23

0.18

0.21

0.24

0.28

0.20

0.25

0.27

T6

0.67

0.51

0.30

0.17

0.70

0.54

0.26

0.18

0.74

0.57

0.32

0.20

T6-a

0.11

0.17

0.03

0.08

0.12

0.22

0.05

0.10

0.13

0.26

0.06

0.12

T6-b

0.36

0.34

0.15

0.15

0.37

0.35

0.16

0.17

0.39

0.37

0.19

0.19

T6-c

0.45

0.55

0.26

0.34

0.59

0.58

0.30

0.37

0.63

0.60

0.32

0.40

T6-d

0.10

0.14

0.11

0.15

0.13

0.17

0.15

0.19

0.14

0.19

0.17

0.22

Average 0.35

0.36

0.19

0.22

0.41

0.40

0.21

0.25

0.43

0.42

0.24

0.28

Table 8 Average over the deck of the shear damage ratio (: ratio of shear force demand to resistance) 0.25 g

0.35 g

0.45 g

Longitudinal

Transverse

Longitudinal

Transverse

Longitudinal

Transverse

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

FBD DBD

C3

0.15

0.14

0.11

0.11

0.16

0.15

0.11

0.11

0.17

0.16

0.11

0.11

C3-a

0.15

0.15

0.11

0.11

0.15

0.15

0.11

0.11

0.16

0.16

0.11

0.12

C3-b

0.14

0.14

0.10

0.10

0.16

0.16

0.10

0.10

0.18

0.16

0.11

0.11

T6

0.05

0.05

0.04

0.04

0.08

0.05

0.04

0.05

0.08

0.05

0.05

0.05

T6-a

0.04

0.04

0.05

0.05

0.05

0.05

0.06

0.06

0.05

0.05

0.07

0.08

T6-b

0.05

0.04

0.04

0.04

0.06

0.04

0.05

0.05

0.06

0.05

0.05

0.05

T6-c

0.08

0.07

0.05

0.05

0.09

0.08

0.05

0.05

0.10

0.08

0.06

0.05

T6-d

0.12

0.14

0.06

0.06

0.15

0.16

0.07

0.07

0.16

0.17

0.08

0.07

Average 0.10

0.10

0.07

0.07

0.11

0.10

0.07

0.07

0.12

0.11

0.08

0.08

of DBD gives a more balanced distribution of inelasticity between the piers and the deck, preventing excessive inelastic action in the latter. Note that the results in Fig. 2 apply for seismic actions from 1.8- to 3.2-times the design one. Besides, the ductility factors refer to deck decompression on the side of the mean tendon and to rebar yielding at the opposite side of the section only. So, the large deck ductility values in Fig. 2 do not imply that the deck actually yields under the design seismic action. The DBD piers have also less (or at most the same) transverse steel than their FBD counterparts. It is interesting, though, to note in Figs. 3 and 5 or Tables 5 and 7 the similar safety margins in flexure or shear in the DBD and FBD bridge versions. These margins are comfortable, even for earthquakes more than 3-times the design seismic action. Regarding the decks in flexure, witness in Fig. 4 the larger safety margin of the decks of DBD bridges under longitudinal earthquake and how uncomfortably close to “ultimate deformation” (i.e., to tendon yielding, or non-prestressed reinforcement rupture, or concrete crushing) some of their FBD counterparts do come. This difference in performance is due to the reduction of inelas-

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Fig. 7 Average (over the 14 longitudinal input motions) flexural damage ratio (: chord rotation demand-toultimate ratio) in bridge T6-c: a–c conventional, force-based design; d–f “displacement-based” design; PGA on rock: a & d 0.25 g; b & e 0.35 g; c & f 0.45 g

tic action in the deck achieved through the present DBD procedure. The decks of the DBD bridges have similar (or slightly better) safety margins in shear as their FBD counterparts. Figures 7, 8, 9, and 10 exemplify the comparison of the performance of the FBD and the DBD versions of one of the eight bridges (a regular, five-span bridge) under the three levels of the ground motion. Figures 7 and 9 in (Bardakis and Fardis 2010) depict also the extent of nonlinear and inelastic action induced in the deck and the piers of these two bridge designs by these motions. Table 9 compares the safety margins provided by the two alternative designs of each bridge, in terms of the value of PGA on rock that exhausts the design criteria of Step 6 in flexure (i.e., a safety factor of 2.0 on the expected ultimate flexural deformation, see Eq. (7)) or of Step 7 in shear (a safety factor of 1.25 on shear resistance, see Eqs. (11), (12)). Overall this DBD approach seems to provide essentially the same safety margin as conventional FBD, despite the much lower reinforcement of its piers.

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557

Fig. 8 Average (over the 7 transverse input motions) flexural damage ratio (: chord rotation demand-to-ultimate ratio) in bridge T6-c: a–c conventional, force-based design; d–f “displacement-based” design; PGA on rock: a & d 0.25 g; b & e 0.35 g; c & f 0.45 g

4 Conclusions A multistep displacement-based seismic design procedure has been presented for concrete bridges having piers integral with a continuous prestressed deck, restrained transversely at the abutments but free to slide there longitudinally. It is well integrated with the usual design practice of the bridge for gravity loads. It entails estimation of pier inelastic deformation demands through 5%-damped modal response spectrum analysis and the “equal displacement” approximation for member deformations, Prescriptive detailing and minimum reinforcement rules are relaxed. Instead, the vertical reinforcement of piers is tailored to seismic deformation demands for uniform ductility ratios across the piers, while their transverse reinforcement is detailed on the basis of explicit checks of their integrity under the design earthquake. The procedure is applied for evaluation purposes to eight bridges having a pre-

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Fig. 9 Average (over the 14 longitudinal input motions) shear damage ratio (: shear force demand-to-resistance ratio) in bridge T6-c: a–c conventional, force-based design; d–f “displacement-based” design; PGA on rock: a & d 0.25 g; b & e 0.35 g; c & f 0.45 g

stressed box girder deck over three or five spans integral with piers of similar or of very different heights. It is found to give more cost-effective and rational designs than the conventional force-based approach of Eurocode 8. Bridges designed with it show an overall improvement in seismic performance under ground motions well beyond the design seismic action, despite having much less reinforcement in their piers.

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Fig. 10 Average (over the 7 transverse input motions) shear damage ratio (: shear force demand-to-resistance ratio) in bridge T6-c: a–c conventional, force-based design; d–f “displacement-based” design; PGA on rock: a & d 0.25 g; b & e 0.35 g; c & f 0.45 g Table 9 Peak Ground Acceleration (PGA) on rock at which the safety margin of 2.0 on flexural capacity or of 1.25 on shear resistance is exhausted anywhere in the bridge

Force-based design (FBD)

Displacement-based design (DBD)

C3

0.30 g

0.38 g

C3-a

0.54 g

0.53 g

C3-b

0.40 g

0.43 g

T6

0.43 g

0.39 g

T6-a

0.43 g

0.39 g

T6-b

0.55 g

0.52 g

T6-c

0.56 g

0.55 g

T6-d

0.48 g

0.49 g

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References Bardakis V (2007) Displacement-based seismic design of concrete bridges. Doctoral Thesis, Civil Eng Dept, Univ. of Patras, Greece Bardakis V, Fardis MN (2010) Nonlinear dynamic v elastic analysis for seismic deformation demands in concrete bridges having deck integral with the piers. Bull Earthq Eng. doi:10.1007/s10518-010-9203-9 Biskinis DE, Fardis MN (2006) Assessment and upgrading of resistance and deformation capacity of RC piers, 1st Europ Conf Earthq Eng & Seismology, Geneva, Paper no.315 Biskinis DE, Fardis MN (2010a) Deformations at flexural yielding of members with continuous or lap-spliced bars. Struct Concret 11(3):127–138 Biskinis DE, Fardis MN (2010b) Flexure-controlled ultimate deformations of members with continuous or lap-spliced bars. Struct Concret 11(2):93–108 Biskinis DE, Roupakias GK, Fardis MN (2004) Degradation of shear strength of RC members with inelastic cyclic displacements. ACI Struct J 101(6):773–783 Caltrans (2006) Seismic design criteria, Version 1.4. California Dept of Transportation, Sacramento Calvi GM, Kingsley GR (1995) Displacement-based seismic design of multi-degree-of-freedom bridge structures. Earthq Eng Struct Dyn 24(9):1247–1266 CEN (2005) European Standard EN 1998-2:2005 Eurocode 8: Design of structures for earthquake resistance Part 2: Bridges. Comite Europeen de Normalisation, Brusells Dwairi H, Kowalsky M (2006) Implementation of inelastic displacement patterns in direct displacement-based design of continuous bridge structures. Earthq Spectra 22(3):631–662 Kowalsky MJ (2002) A displacement-based approach for the seismic design of continuous concrete bridges. Earthq Eng Struct Dyn 31(3):719–747 Kowalsky MJ, Priestley MJN, MacRae GA (1995) Displacement-based design of RC bridge columns in seismic regions. Earthq Eng Struct Dyn 24(12):1623–1643 Panagiotakos TB, Fardis MN (1999) Deformation-controlled earthquake resistant design of RC buildings. J Earthq Eng 3(4):495–518 Panagiotakos TB, Fardis MN (2001) A displacement-based seismic design procedure of RC buildings and comparison with EC8. Earthq Eng Struct Dyn 30:1439–1462 Priestley MJN (1995) Myths and fallacies in earthquake engineering—conflicts between design and reality, Thomas Paulay Symposium: Recent Developments in Lateral Force Transfer in Buildings, ACI SP-157, Detroit, Mi Priestley MJN, Calvi GM (1997) Concepts and procedures for direct displacement-based design and assessment. In: Fajfar P, Krawinkler H (eds) Seismic design methodologies for the next generation of codes. Balkema, Rotterdam pp 171–182 Priestley MJN, Calvi GM, Kowalsky MJ (2007) Displacement-based seismic design of structures. IUSS Press, Pavia SEAOC (1999) Recommended Lateral Force Requirements and Commentary. Seismology Committee, Structural Engineers Association of California, Sacramento Shibata A, Sozen M (1976) Substitute structure method for seismic design in reinforced concrete. J Struct Eng ASCE 102(1):1–18

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