s2-1_formatted ddbd paper

8/7/2019 S2-1_Formatted DDBD paper

1/12

DISPLACEMENT-BASED SEISMIC DESIGN OF BRIDGES

NIGEL PRIESTLEYCentre of Research and Graduate Studies in Earthquake Engineering and Engineering Seismology (Rose School),Istituto Universitario di Studi Superiori (IUSS) Pavia, Italy

MICHELE CALVIDepartment of Structural Mechanics, Univerita degli Studi di Pavia, Pavia Italy

Abstract: A recently completed research project on displacement-based seismic designof structures has included emphasis on bridge design. This papersummarizes the work carried out in this project. A brief discussion ofconceptual problems with force-based seismic design of concrete bridges isfollowed by a description of displacement-based seismic designfundamentals. Seismic input for displacement-based design is brieflydiscussed, leading to examination of characteristics of bridges which couldbe expected to respond elastically to the design-level seismic input.Methodologies for longitudinal and transverse seismic design of irregular

multi-span bridges are presented, and a brief discussion of capacity-designeffects for bridges is included.

Key words: displacement, transverse, longitudinal, design, isolation

1. INTRODUCTION:PROBLEMS WITH FORCE-BASED SEISMIC DESIGNConventional seismic design of bridges is force-based. That is, force-levels correspondingto elastic response to a design acceleration response spectrum are calculated based onelastic stiffness estimates. These elastic force levels are then divided by a force-reductionfactor representing the assessed displacement ductility capacity. The structure is thendesigned for these reduced force levels, and the displacement may be checked to ensurethat code-specified drift limits are not exceeded. Problems with this procedure arediscussed in detail in Priestley et al [2007], and will only be briefly summarized here:

Elastic stiffness (1). This is not known at the start of the design process, and veryapproximate values are used in design. Since the bridge periods, and hence thedesign base shear depend on the elastic stiffness, significant errors in the daseshear force can be expected.

Elastic stiffness (2). The distribution of base shear strength between piers on thebasis of elastic stiffness is illogical, and is based on the assumption that piers can


2/12

Nigel Priestley and Michele Calvi2

be forced to yield at the same displacement, despite having different stiffnesss.For example if the bridge is supported by two piers of height 5m and 10mrespectively, then the shear force will be distributed between the columns in theratio 1: 0.125, resulting in base moment ratios of 1.0 to 0.25. This is illogical.

Foundation effects are generally ignored in force-based design, and are difficultto incorporate in the design process as they affect both elastic period, anddisplacement ductility demand.

Dual load paths: The distribution of required strength under transverse response,between elastic superstructure flexure and inelastic pier flexure cannot be basedon a logical framework in force-based design.

Displacement-equivalence rules relating inelastic to elastic displacement demandare based on inappropriate time-history analyses, as discussed in detail in anotherpaper at this workshop (Priestley et al, 2007b).

Please read carefully the instructions presented herein for the format of a ProceedingBook to be printed by the IUSS Press. This document has been written in the requiredformat and so please study the document for any formatting requirements that are not

explicitly stated in the text. Any formatting styles not described herein are left to theauthors discretion.

2. FUNDAMENTALS OF DIRECT DISPLACEMENT-BASED DESIGN (DDBD) The problems outlined above disappear when design is based on the substitutestructureanalysis approach initially developed by Gulkan and Sozen [1974] and Shibataand Sozen [1976], and developed into a design approach in Priestley et al [2007]. Sincethe basic principles have been outlined in previous papers (e.g. Priestley and Calvi, 2003)they will be only briefly reviewed here, with reference to Fig.1 which considers a SDOFrepresentation of a bridge.

While force-based seismic design characterizes a structure in terms of elastic, pre-yield,properties (initial stiffness Ki, elastic damping), DDBD characterizes the structure bysecant stiffness Ke at maximum displacement d (Fig.1(b)), and a level of equivalent

viscous damping, representative of the combined elastic damping and the hystereticenergy absorbed during inelastic response.

With the design displacement at maximum response determined as discussedsubsequently, and the corresponding damping estimated from the expected ductilitydemand, the effective period Te at maximum displacement response, measured at the


3/12

1st

US-Italy Seismic Bridge Workshop 3

effective height He (Fig.5(a)) can be read from a set of displacement spectra fordifferent levels of damping, as shown in the example of Fig.1(d).

me

0 2 4 6Displacement Ductility

0

0.05

0.1

0.15

0.2

0.25

Damping(%)

0 1 2 3 4 5Period (seconds)

0

0.1

0.2

0.3

0.4

0.5

Displacement(m)

5%

10%

15%

20%

30%

d

Te

Elasto-Plastic

Steel Frame

Concrete Frame

Hybrid Prestress

(c) Equivalent damping vs. ductility (d) Design Displacement Spectra

Concrete Bridge

Figure 1 Fundamentals of Direct Displacement-Based Design

The effective stiffness Ke of the equivalent SDOF system at maximum displacement canbe found by inverting the normal equation for the period of a SDOF oscillator to provide

22/4 eee TmK = (1)

F

he

Fu

Fn

Ke

Ki

y d(a) SDOF Simulation (b) Effective Stiffness Ke


4/12


where me is the effective mass of the structure participating in the fundamental mode ofvibration. From Fig.1(b), the design lateral force, which is also the design base shear forceis thus

(2)deBaseKVF ==

2.1 Design DisplacementThe design displacement of the equivalent SDOF structure (the generalized displacement

coordinate) is based on the inelastic mode shape for the bridge considered, and is givenby

(3)( ) ( )==

=n

i

ii

n

i

iidmm

11

2/

where mi and i are the masses and displacements of the n significant mass locationsrespectively. The design procedure will identify the displacement capacity c of thecritical element, normally, though not exclusively, the shortest pier, based on limit strainsapplicable for the limit state considered.

For longitudinal response of a straight bridge the displacements of all superstructure

masses will generally be identical, unless the bridge is very long, and axial deformationeffects are significant. For transverse response, the mode shape will depend on relativetransverse stiffness of superstructure and piers, presence or absence of internalmovement joints, and degree of fixity at supports. Some possible mode shapes areillustrated in Fig.2.

2.2 Effective MassThe effective mass participating in the fundamental inelastic mode is given by

(4)( ) dn

i

iie mm ==

/

1

2.3 Equivalent Viscous DampingFor bridges, the equivalent viscous damping for each individual element depends on thelocal displacement ductility and is given by

+=

1444.005.0 (5)


5/12

1st


(a) Symm., Free abuts. (b) Asymm., Free abuts. (c) Symm., free abuts.Rigid SS translation Rigid SS translation+rotation Flexible SS

2

3

4

5

1

2

2 2 2

1 1 1

3

3 3 3

4

4 4 4

5

5 55

(d) Symm,. Restrained abuts. (e) Internal movement joint (f) Free abuts., M.jointFlexible SS Rigid SS, Restrained abuts. Flexible SS

1

Figure 2 Different Possible Transverse Displacement Profiles for Bridges

The system damping is then found as an average of the damping of the individualelements, weighted by shear force and displacement. For longitudinal response the designdisplacements will be equal, and hence:

=

mi

m

ii

sys

V

V

(6)

2.4 The Design ProcessThe design process is then straightforward. For longitudinal design, the total base shearforce is found from Eq.(2), and distributed to the piers. The way in which thisdistribution is effected is a designers choice, but will normally be based on theassumption of equal moment capacity (and hence identical reinforcenment details) at thebase of all piers. Note that this is markedly different from force-based design, where the


6/12


design moments would be inversely proportional to the pier heights squared. If thedecision to have equal moment capacity at the base of all piers is made at the start of thedesign process, then the relative proportions of shear force carried by the piers is known,and Eq.(6) can be solved even though the absolute magnitudes of pier shears are stillunknown.

For transverse design the situation is more complex. This is discussed with reference toFig.3.

displaced shape

V1 V52 3 41 5

Figure 3 Components Contributing to Damping under Transverse Response

In this case a proportion of the base shear force is carried by elastic superstructure flexureback to the abutments, which may be flexible, with significant damping. A general systemdamping expression for the 4-span bridge in Fig.3 can be written as

( ) ( )

( ) ( )

==

==

++

++

=4

2

4

2

4

2

4

2

1/

11

1/

11

i ii

i

i

aad

i ii

ii

i

aaSSad

e

HHxxx

HHxxx

(7)

where a is the average abutment displacement, a and SS are the damping ratiosappropriate for abutment and superstructure displacements, and x is the proportion oftotal base shear transmitted to the abutments by superstructure flexure. Since thisproportion will be unknown at the start of the design process, some iteration is involved.

An inelastic mode shape, and a value for x are assumed. The design displacement andthe equivalent viscous damping are calculated, and hence the base shear computed. Theeffective stiffness of the piers corresponding to the assumptions for x and the designdisplacement profile can thus be found. A structural analysis based on the assumed

F1 F2 F3 F4 F5

V2V3

V4H2 H3

H4


7/12

1st


stiffness distribution is carried out, and the displacements checked. The value for x, andthe displacement profile are modified by successive approximations to provide agreementbetween the design profile and the analysius results. In practice this process is found toconverge rapidly. Full details of the procedure are available in Priestley et al [2007].

2.5 Capacity Design IssuesCapcity design is the terminology given to the process of ensuring that undesireablemodes of inelastic deformation, such as shear failure, do not occur. In bridge design there

are three aspects that require consideration:

Pier shear forces may be higher than those corresponding to the design forcedistribution as a consequence of pier flexural strength exceeding design values, oras a consequence of higher mode response resulting from inertial mass of thepier, or torsional mass inertia of the superstructure.

Higher mode response of the superstructure may result in increasedsuperstructure seismic moments, compared with values corresponding to thedesign lateral force distribution.

Abutment reactions may also be influenced by superstructure higher moderesponse. In fact, this is the most critical aspect of higher mode response.

It has been found during development of DDBD methodology for bridges that highermode effects are best represented by results from a modified modal superpositionapproach, where the stiffness of members responding inelastically (e.g. the piers) isrepresented by the effective secant stiffness to maximum displacement response. Fulldetails are available in Priestley et al [2007].

3. DDBD INCLUDING SEISMIC ISOLATIONAdaptation of the design procedure to incorporate seismic isolation is straightforward.

The design objective is first set. With reference to Fig.3, this could be to achieve uniformtransverse displacement at the superstructure at the top of each pier and abutment, by theaddition of seismic isolation devices between the piers and superstructure, and to usecapacity design procedures to ensure that the piers remain elastic under the design levelof excitation. In this case the system design displacement will be equal to thesuperstructure displacement, and can be made equal in both longitudinal and transversedirections. It will also normally be based on the isolator displacement capacity, and maybe equated to the plateau displacement of the displacement response spectrum, for thesystem level of damping. This results in a minimum-strength design.


8/12


The effective damping of each pier/isolator combination can be found from

isopier

isoisopierpier

+

+=

(8)

where the displacement of the pier, pier is reduced from the pier yield displacement byabout 20% to allow for possible overstrength in the isolator devices. This is essentially acapcity design measure.

The proportional distribution of base shear between piers can be decided at the start ofthe design process, and could be chosen to have uniform isolator properties, or to haveuniform piers strengths, for example. Again, full design details are presented in Priestleyet al [2007].

4. DESIGN EXAMPLE: FOUR-SPAN BRIDGE UNDER TRANSVERSE EXCITATION The four-span bridge represented in Fig.4 is designed by DDBD principles, first forconventional seismic resistance and second incorporating seismic isolation.

40m 50m 50m 40m

12m

Figure 4 Bridge Design Example

Superstructure depth is 2m, and superstructure mass averages 190kN/m including weightinternal cap beams. The superstructure is torsionally flexible. Columns are 2.4m indiameter. The design displacement response spectum corresponds to the ATC32 [1996]spectrum for medium ground and a PGA of 0.6g. Expected concrete compressionstrength is 39MPa, and yield strength is 462MPa. The superstructure is restrained laterallyby shear keys at the abutments, and abutment displacements are to be limited to 40mm.

Transverse reinforcement is chosen as D20 @ 100mm crs, which it is found results in adamage-control limit-state curvature at the pier bases of 0.0283/m.

Clearly the displacements of the central pier and the abutments are the key designconstraints. The limit state displacement at the mid-height of the superstructure is foundto be 0.596m, and the initial estimate of the displacement at piers B and D is 0.7x0.596 =

16m 16mA E

C

(Not to scale)DB


9/12

1st


0.417m. A further initial assumption is that 50% of the lateral inertia force is carried backto the abutments. With these assumptions the design SDOF displacement from Eq.(3) isfound to be 0.485m, and the equivalent viscous damping ratio from Eq.(7) is 0.089.

Table 1. Final Data for Ductile Design of Bridge Example of Fig.4

Step 3 d = 0.477mStep 4 me = 2834 tonnesStep 5 = 1.52 (Piers B and D); = 3.83 (Pier C)

= 0.098 (Piers B and D); = 0.154 (Pier C)Step 6

=

=5

1

139.0

i

iFV (Piers B and D); (Pier C)=

=5

1

182.0

i

iFV

Step 7 e = 0.085Step 8 Te= 2.20 sec; Ke = 23191 kN/m; VBase = 11061 kNStep 9 FA= FE= 127 kN; FB = FD= 2969 kN; FC= 4869 kNStep 10 VA= VE= 2987 kN; VB= VD = 1538 kN; VC= 2012 kN

KA = KE= 74664 kN/m; KB = KD= 3865 kN/m; KC= 3375 kN/mStep 11 A = E= 0.039m; B = D= 0.406m; C = 0.613mStep 12 x= 0.51

VA= VE= 2821 kN; VB = VD= 1639 kN; VC= 2143 kN

KA= KE= 70516 kN/m; KB = KD= 4117 kN/m; KC= 3595 kN/m

A = E= 0.040m; B = D= 0.395m; C= 0.595m The iterative procedure described in Section 2.4 above requires only two iterations todevelop a stable solution which is summarized in Table 1. Note that this solution is afundamental mode solution, and does not include consideration of higher modeeffects.

In order to verify the design displacements, and to investigate higher mode effects particularly on the abutment response an inelastic time-history analysis is carried outusing three spectrum-compatible accelerograms modified from real accelerograms using

wavelet theory. The structure was represented as a 3-D structure, with the superstructurerepresented by an elastic member, while the piers used Thin Takeda hysteretic

characteristics with a 5% second slope stiffness. Elastic damping was represented by 5%tangent-stiffness damping. The abutments were represented by lateral springs withdifferent properties in different analyses, as discussed subsequently. Each span wasdivided into five segments, with tributary mass distributed to the superstructure nodes.

Results, in terms of response displacements and superstructure moment envelopes arecompared with the design profiles in Fig.5(a). The design profiles are shown as a solidbold line, and the results from the individual records by light lines. The average of thethree analyses is shown as a dashed bold line. It will be noted that the agreement between


10/12


the design and analyses profiles is very good, and that the scatter between the results ofthe records is small.

0 40 80 120 160Distance (m)

0

0.2

0.4

0.6

Displacement,(m)

0 40 80 120 160Distance (m)

0

40

80

120

160

200

Moment(MNm)DesignITHA, ave

DesignITHA, ave

0 40 80 120 160Distance (m)

0

0.2

0.4

0.6

Displacement(m)

0 40 80 120 160Distance (m)

0

40

80

120

160

200

Mom

ent(MNm)

0 40 80 120 160Distance (m)

0

0.2

0.4

0.6

Displacement(m)

0 40 80 120 160Distance (m)

0

40

80

120

160

200

Moment(MNm)

DesignITHA ave

Design

ITHA ave

DesignITHA

Design

ITHA

(a) Design, Elastic Abutments(Displacement envelopes; Moment envelopes)

(b) Design, Elasto-plastic abutments (Displacement envelopes; Moment envelopes)

(c) Revised, Elasto-plastic abutments (Displacement envelopes; Moment envelopes)

Figure 5 Displacement and Moment Profiles from Time-History Analysis of DuctileDesign


11/12

1st


The average central displacement is 4% larger than the design value of 0.595m, and peaksuperstructure moments are very close to the design values. However, though not veryapparent in Fig.5(a), the displacements at the abutments, at an average of 68.5mm are71% higher than the design value of 40mm. It might be felt that this is an acceptableexcess, as it only represents a small ductility demand. To provide further investigation ofthis, it is decided to rerun the analyses, with the abutments modelled as elasto-plasticsprings. Results from this set of analyses are shown in Fig.5(b), where for clarity only theaverage of the three analyses is compared with the design values. The peak displacementat the central pier is essentially unaffected (the excess is reduced to 3%), but the lateral

displacements at the abutments are increased to 118mm (4.6in). This increase above thedesign value is due to higher mode effects, as explained in the previous section. It mightstill be considered an acceptable result, but we decide to attempt to reduce thedisplacements to the design level. As discussed in the previous section, this could bebased either on an EMS analysis or on time-history results. Since we have the results ofthe time-history analysis, we use them directly.

The initial elastic analyses indicated that the design forces in the abutments, includinghigher mode effects, were 71% higher than the design levels. This implies that thestiffness and strength of the abutments should be increased by a dynamic amplificationfactor of at least wa= 1.71 if the displacements are to be constrained to the design levelof 40mm. However, increasing the stiffness is likely to further increase the design forces,so, based on experience we chose to increase the abutment stiffness and strength by100%.

A new set of analyses were run based on this assumption, (and with the abutmentsmodelled as E-P elements) and the results are plotted in Fig.5(c). The displacementprofile is now almost perfect, with a peak central displacement less than 2% above thedesign value, and the average abutment displacement is 37mm (1.5in), indicating elasticresponse. Full details of the design and analyses are available in Priestley et al [2007].

The bridge is now redesigned, using friction pendulum isolators, whose equivalent viscous dampling ratio is estimated at 20%. It is decided to aim for a uniform

superstructure transverse displacement (also at the abutments) of 0.5m. The piers are tobe designed such that at expected isolator response force levels, the pier yielddisplacements ar no more than 80% of yield displacement. The base shear force isdistributed to the piers in proportion to gravity load, though other options would beequally viable. With these decisions the following design shears are obtained:

VA= VE= 0.622MN

VB= VD= 1.40MN

VC= 1.56MN


12/12


It will be noted that the shear forcesare significantly lower than for the ductile design,particularly at the abutments, which would allow for significant reduction in abutmentdesign costs.

Figs 6 and 7 show the results of time-history verification with design values fordisplacement and isolator shear force. Agreement is excellent.

0.0

0.2

0.4

0.6

0.8

0 50 100 150

location (m)

Deckdisplace

ment(m)

Average TH

Design

Figure 6 Design and AverageTime-History Displacements for Isolated Bridge

0

500

1000

1500

2000

-10 40 90 140 190

location (m)

Shear(kN

)

Average TH

Design

Figure 7. Design and Time-History Isolator Forces for Isolated Bridge

5.

REFERENCESGulkan, P. and Sozen, M. (1974) Inelastic Response of Reinforced Concrete Structures to

Earthquake Motions. ACI Journal, Vol 71(12) pp 604-610)

Shibata, A. and Sozen, M. (1976) Substitute Structure Method for Seismic Design in R.C. ASCE

Journal of Structural Engineering, 102(1) pp1-18

Priestley, M.J.N. and Calvi, G.M. (2003) Direct Displacement-Based Seismic Design of Bridges

Proc. ACI Special Seminar on Seismic Design of Bridges, San Diego,

Priestley, M.J.N., Calvi, G.M., and Kowalsky, M.J. (2007) Displacement-Based Seismic Design of

Structures IUSS Press, Pavia, 721pp.

s2-1_formatted ddbd paper

Documents