Bridge support elastic reactions under vertical earthquake ground motion

Engineering Structures 31 (2009) 2317–2326

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Engineering Structures

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Bridge support elastic reactions under vertical earthquake ground motionFrederic Legeron a,1, M. Neaz Sheikh b,∗a Department of Civil Engineering, University of Sherbrooke, Sherbrooke (Quebec), Canada J1K 2R1b School of Civil, Mining and Environmental Engineering, University of Wollongong, NSW 2522, Australia

a r t i c l e i n f o

Article history:Received 3 October 2008Received in revised form1 May 2009Accepted 1 May 2009Available online 30 May 2009

Keywords:BridgeSeismic designVertical earthquake ground motionSupport reactionSimplified method

a b s t r a c t

In North America, bridges are not typically designed for vertical earthquake groundmotion. Design codes,AASHTO in the USA and S6-06 in Canada, do not explicitly consider the effect of vertical earthquakeground motion in the design of highway bridges. However, analytical and field evidences have oftendrawn engineers’ attention to the damage potential of vertical earthquake ground motion to engineeredbridges. This paper proposes a simplified method to calculate elastic support reactions under verticalearthquake ground motion. Support reactions are first calculated by exact dynamic method. By applyinga few assumptions, a simplified method has been developed. The developed method can be readilyused in the design of typical bridges with regular span distribution. This simplified method has beendemonstrated with several examples.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Vertical earthquake ground motion is not explicitly consideredin American and Canadian design codes for bridges [1,2]. CanadianHighway Bridge Design Code [2] takes into account the verticalearthquake ground motion in a simplified way by increasing anddecreasing the dead load action in load combinations, irrespectiveof earthquake magnitude, fault distance, and site soil condition.In the Caltrans seismic design criteria [3], vertical ground motionis only considered as an equivalent static vertical load when thesite peak rock acceleration is 0.6g or greater. The recent edition ofEurocode [4] has proposed to take into account the effect of verticalearthquake groundmotion when the bridge is located within 5 kmof an active seismotectonic fault or in high seismic zones.When the effects of vertical earthquake ground motion are

explicitly considered in the design process, the vertical responsespectrum is taken typically as the two-thirds of the horizontalresponse spectrum for the entire period range of engineeringinterest. This approach was originally proposed by Newmarket al. [5] and has since been widely used. However, analysesof strong motion data indicate that in the vicinity of moderateto strong motion earthquakes, vertical to horizontal (V/H) peakground acceleration ratio often exceeds unity; hence, the 2/3rule is not reasonable [6–10]. In addition, a recent research

∗ Corresponding author. Tel.: +61 2 4221 3009; fax: +61 2 4221 3238.E-mail addresses: [email protected] (F. Legeron),

[email protected] (M. Neaz Sheikh).1 Tel.: +1 819 821 7395; fax: +1 819 821 7974.

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.05.001

investigation has also shown that V/H spectral ratio can approach1.8 at large magnitude earthquake events for short site sourcedistances and short periods [10]. Some recent studies have focusedon the shape of the vertical response spectrum and proposeddesign vertical acceleration response spectrum consisting of a flatportion at short periods (0.05–0.15 s) and a decaying spectralacceleration at longer periods [8,10,11]. Typically, the verticalground motion has lower energy content than the horizontalground motion over the frequency range of interest. However,all its energy is concentrated in a narrow frequency band andcan be destructive to the engineered structures having verticalfrequencies within that range (5–20 Hz).A literature survey indicates that only a few studies have been

conducted to quantify the effect of vertical ground motion onbridges [12–15]. Analytical studies, in line with field observations,found that certain failure modes are directly related to highvertical ground acceleration [15]. In addition to the possibility ofcompressive over-stressing or failure due to direct compression,vertical motion may induce failure in shear and flexure andcan cause bearing and expansion joint failures. All the analyticalstudies carried out to date are based on finite element modellingof the bridges and emphasize the importance of incorporating thevertical ground motion in the design process [12–15]. However,these studies have not clearly defined the way to incorporate theeffects of vertical ground motion into the design process. One ofthe studies emphasized the need to develop a simplified methodfor the inclusion of the vertical earthquake ground motion in thedesign process [15].The purpose of the current study is to develop a simplified

method to quantify the elastic support reactions of bridges under

2318 F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326

Nomenclature

The following notations are used in this paper:

A Zonal acceleration ratioa Response spectral acceleration at the effective

period of supports (for rigid bridge: maximumresponses spectral acceleration)

Csm Elastic seismic coefficientEI Stiffness of superstructureI Importance factorL Main span lengthLn Modal participation factor of mode nMn Modal mass of mode nRi Support reaction of support iSa Response spectral accelerationSa,max Maximum response spectral accelerationSd Response spectral displacementTn Modal period of the bridgeTeff Effective natural period of the supportw Superstructure unit weightζ Scaling factorϕn Mode shape of mode nφ Phase angleψ Ratio of length between end span and main spanΩn Spatial natural frequencyΓ n Unit modal support reaction of mode nµ Superstructure unit mass

vertical earthquake groundmotion. It is noted that nonlinear effectis not generally considered for the vertical earthquake groundmotion [4]. The developed method requires few calculations andcan be readily used by design offices for the design of simplehighway bridges and preliminary design of more complex bridges.The method has been found to be in excellent agreement with theanalytical results.

2. Calculations of support reactions under vertical earthquakeground motion

Generally, threemethods are used in practice for the calculationof support reactions: (i) Rayleigh method: This method has beenrecommended by American and Canadian bridge design codes [1,2] for irregular ordinary multispan bridges and regular essentialor emergency-route bridges in seismic performance zone 2 or less.Rayleighmethod is generally not recommended for vertical groundmotion since the choice of deflection shape for a complex systemis not straightforward [16] and does not provide results close tothe exact calculation in most cases (refer examples in Section 5).(ii) Modal analysis: This method is most suitable for structureswith irregular geometry, mass and stiffness. This method doesnot take into account the nonlinear effects. However, nonlineareffect is not considered significant for vertical earthquake groundmotion [4], as mentioned earlier. Moreover, ductility of bridgepiers under vertical earthquake ground motion is not well knownand usually considered to be low. (iii) Time history analysis: Thismethod is very complex and time consuming and is usually carriedout only for critical structures in high seismic zones [1]. Moreover,the choice of representative ground motion may add complexityin applying this method. Hence, elastic modal analysis has beenconsidered sufficient for the calculation of support reactions undervertical earthquake ground motion in this study.It is also possible to calculate the support reactions directly by

applying dynamics of continuum [17]. Considering an unloaded

Fig. 1. Equilibrium of a slice of bridge superstructure.

slice of bridge superstructure (Fig. 1) and assuming superstructureunit mass µ(x) = µ and stiffness EI(x) = EI , equilibrium of forcesresults in:

∂2z(x, t)∂t2

+EIµ

∂4z(x, t)∂x4

= 0. (1)

Solution of this equation can be found by classical separation ofspatial and time variables:

Z(x, t) = Z(t) · ϕ(x)= (A cos(Ωx)+ B sin(Ωx)+ C cosh(Ωx)+D sinh(Ωx))× cos(ωt + φ) (2)

where A, B, C and D are integration constants related to boundaryconditions; φ is a constant depending on initial conditions; Ω isthe spatial natural frequency; and ω is the natural frequency. Theyare related by Eq. (3).

Ω2 = ω

õ

EI. (3)

There are four unknowns in each span (A, B, C and D). Foreach span, two equations can be written for zero displacement atsupports and another two equations can be written for continuityofmoments (or zeromoments at end support). It results in a systemof 4r equations where r is the number of spans. Using matrixnotation:

[M] EX = E0 (4)

where [M] is a real matrix of 4r × 4r-dimension with terms± cos(LiφΩ), sin(LiφΩ), cosh(LiφΩ) and sinh(LiφΩ), and Li is thelength of span i.EX is a vector of 4r-dimensionswith components A1, B1, C1,D1,

. . . , Ai, Bi, Ci,Di, . . . , Ar , Br , Cr ,Dr.The system of 4r equations has nontrivial solution only if

Det(M) = 0 (5)

which has 4r eigenvaluesΩn corresponding to eigenmodes ϕn(x).Eigenmodes are chosen so that

∫ϕn(x)dx = 1 to create an

orthogonal base of eigenvectors.This set of equations can be solved for some typical bridges. The

bridge geometry is chosen as follows:(i) single-span bridge with span length L;(ii) two-span bridge with first span length Ψ L and second spanlength L;

(iii) three-span bridge with end spans length Ψ L and central spanlength L;

(iv) four-span bridge with end spans lengthΨ L and interior spanslength L.

Ωn can be calculated as a function of 1/L according to Eq. (6):

Ωn =αn,ψ

L. (6)

F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326 2319

Table 1Values of αn,ψ .

Bridge type ψ n1 2 3 4 5 6

Single-span 1.0 3.1415 6.2832 9.4248 12.5664 15.7080 18.8496Two-span 0.5 3.5560 6.2815 7.4273 9.8479 12.5680 13.7120

0.6 3.5061 5.6982 6.9240 9.6741 11.1850 13.03600.7 3.4498 5.0597 6.7450 9.1990 10.1660 12.81000.8 3.3785 4.5499 6.6399 8.3275 9.8731 12.09100.9 3.2794 4.1768 6.5148 7.5698 9.7231 10.98901.0 3.1416 3.9266 6.2832 7.0686 9.4248 10.2100

Three-span 0.5 3.9266 6.2832 7.0685 7.8532 10.2100 12.56600.6 3.8159 5.5822 6.0508 7.3515 9.8550 11.00600.7 3.6891 4.9152 5.3370 7.1180 9.1362 9.61240.8 3.5343 4.3659 4.8442 6.9133 8.2081 8.57170.9 3.3477 3.9208 4.5147 6.6553 7.3872 7.85371.0 3.1416 3.5564 4.2975 6.2832 6.7076 7.4295

Four-span 0.5 3.5564 4.2975 6.2832 6.7076 7.4295 7.85320.6 3.5060 4.2209 5.6983 5.8918 6.9143 7.56860.7 3.4497 4.1089 5.0597 5.2393 6.7450 7.44260.8 3.3755 3.9268 4.5499 4.8162 6.6405 7.30560.9 3.2656 3.6707 4.1768 4.5860 6.5148 7.02551.0 3.1416 3.3932 3.9266 4.4633 6.2832 6.5454

Values of αn,ψ for the first six vertical modes are presented inTable 1. FromΩn, the natural periods or frequencies of the bridgecan be determined:

T n,ψ =2π(αn,ψ

)2õL4

EI. (7)

Support reaction is calculated for each mode as:

Rni = EI

(d3ϕn

(x+i)

dx3−d3ϕn

(x−i)

dx3

)Ln

MnSd(ωn, ξ n

)= EI

[d3ϕn (x)dx3

]x+ix−i

Ln

MnSd(ωn, ξ n

)(8)

with Ln =∫structure ϕ

n (x) µdx is the modal participation factor ofmode n; Mn =

∫structure [ϕ

n (x)]2 µdx is the modal mass of moden; Sd (ωn, ξ n) is the maximum displacement equal to the responsespectral displacement of mode n and Sa is the response spectralacceleration of mode n.This can be rewritten as:

Rni = µL×1

Ωn4L

[d3ϕn (x)dx3

]x+ix−i

Ln

Mn× Sa

(ωn, ξ n

)= µLΓ ni Sa

(ωn, ξ n

)(9)

with the unit modal support reaction as:

Γ ni =1

Ωn4L

[d3ϕn (x)dx3

]x+ix−i

Ln

Mn. (10)

It is important to note that Γ n is constant for all bridges of sameclass—a class being defined as same number of spans and samespan arrangement. To illustrate the notion of bridge class, twobridgeswith same span arrangement are considered: the referencebridge with main span L and a scaled bridge with main span ζ L.If mode n and spatial pulsation of the mode are ϕn(x) andΩn res-pectively for the reference bridge, themode n and spatial pulsationfor the scaled bridge are ϕn′ (ζ x) = ϕn (x) and Ωn = ζΩn′. Theunit modal reaction for the scaled bridge is:

Γ n′=

1Ωn′4ζ L

[d3ϕn′ (x)dx3

]x+ix−i

Ln′

Mn′

=1

Ωn4

ζ 4ζ L

[1ζ 3

d3ϕn (x)dx3

]x+ix−i

ζµ′

µLn

ζµ′

µMn

=1

Ωn4L

[d3ϕn (x)dx3

]x+ix−i

Ln

Mn= Γ n. (11)

This result is of prime importance since it is now possible toperform calculations on a reduced number of reference bridges andextend the results to all other bridges. Γ n is independent of spanlength L, unit mass of bridge µ, and stiffness of bridge. It is onlydependent on the class of bridge.The reaction at support i can be computed as:

Ri =√ ∑n=1...4n

Rn2i = µL√ ∑n=1...4r

S2a (ωn, ξ n)Γn2i . (12)

It is possible to calculate support reactions from Eq. (12). However,the calculation procedure is complicated and may not be practicalfor practicing engineers. Section 3 described the development ofsimplified methods for calculating support reactions.

3. Simplified method for support reactions

As mentioned earlier, calculating support reactions fromEq. (12) is tedious; however, the equation can be simplifiedassuming the bridges as rigid or flexible.

3.1. Simplified method for rigid bridges

Most typical short andmedium span bridges are very stiff in thevertical direction. Therefore, the firstmodeswhich contributemostto the seismic response have periods below the period at whichdesign acceleration response spectrum starts to descend (refer toSection 4). Hence, Sa can be considered as the maximum responsespectral acceleration (Sa,max) for rigid bridges. This will provide aslightly conservative estimate of the support reactions, as someof the contributing higher modes, though not significant, may fallin the initial ascending branch of the response spectrum (period<0.05 s) when such an ascending branch exists in Code. It is notedthat Sa,max has been designated as a (in terms of g) for rigid bridgesherein. From Eq. (12), the support reaction can be rewritten as:

Ri =√ ∑n=1...4n

Rn2i = µgLa√ ∑n=1...4r

Γ n2i . (13)

2320 F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326

Fig. 2. Values of parameter b.

Considering bi =√∑

n=1...4r Γn2i , Eq. (13) can be rewritten as

Ri = abiwL (14)wherew is the unit weight of bridge superstructure (w = µg).It was proven that Γ n is dependent only on the class of bridge

Eq. (11). Hence, biwill also be dependent only on the class of bridge.Fig. 2 presents bi-values for each class of bridge. The bi-valueshave been calculated from the theoretical approach developed inSection 2. The computer program MAPLE 10 [18] has been usedto calculate mode shapes and vibration frequencies for the firstsix vertical modes. Modal combinations have been carried outby Square-Root-of-Sum-of-Squares (SRSS) method, since verticalmodes are well separated. In order to verify the results, finiteelement computer program SAP 2000 [19] has been used for theanalysis using a unit spectrum (Sa(w, ζ ) = 1.0) for up to 25verticalmodes. The results have been found in excellent agreementwith each other. In Fig. 2, b-values have been presented only forbridges up to 4 spans. This is mainly because it is not very commonin practice to design bridges with continuous superstructure over4 spans.However, the proposed method can be extended for bridges of

more than 4 spans. Numerical calculations have shown that it ispossible to extend the results from 4 spans to n spans by taking 1.1times R3 for Ri with i = 4 or more (bridges of 4 or more spans).For example for a seven-span bridge, support reactions for eightsupports would be R1, R2, R3, 1.1R3, 1.1R3, R3, R2, R1, respectively,where R1, R2 and R3 are the reactions obtained with the b-valueof a four-span bridge of similar arrangement. Computer programSAP 2000 has been used to analyze bridges up to 12 spans to verifythe proposed method. The predicted values have been found inreasonable agreement with the values obtained from SAP 2000analysis (Fig. 3).

3.2. Simplified method for flexible bridges

For flexible bridges, response spectral acceleration is lowerthan maximum spectral acceleration, a, as the periods of theflexible bridges may fall in the descending branch of the spectralacceleration [1,2,4]. Hence, calculation of support reactions usinga-value as the maximum spectral acceleration would be overlyconservative.As a simplification of Eq. (12), it is proposed to calculate the

support reaction using Eq. (14), after identifying the parameter a,

R1

R2

R3

R4

R5

R6

R7

0.2

0.4

0.6

0.8

bca

lcul

ated

0.0

1.0

0.2 0.4 0.6 0.8

b predicted

0.0 1.0

Fig. 3. Predicted versus calculated b-value.

which reflects the fact thatmain contributingmodes are not on thedesign spectra plateau but on the descending branch. This can bedone easily if an effective period of each support, Teff (i,ψ), can becalculated independently. Extensive numerical calculations werecarried out for various bridges. For each bridge, support reactionsRi,ψ are calculated with the exact method developed in Section 2.From thismethod the value of a for each support of a class of bridgeis computed. The corresponding effective period of the supportTeff (i,ψ) is calculated that would provide the appropriate value ofa with the spectra used in the calculation for which a = Sa(Teff ).The value of the effective period of each support is seen to dependonly on the class of bridge and reaction under consideration andcan be expressed as:

Teff (i,ψ) =2πα2i,ψ

õL4

EI. (15)

Values of αi,ψ are presented in Table 2. The method presentedherein to calculate Teff (i,ψ) is related to the input responsespectrum, but the influence of the spectra is in fact very limitedas the effective period is usually very close from the modethat provides the highest contribution to the reaction underconsideration or a weighted average of the two periods providingthe most contribution.

4. Code recommendations for vertical earthquake groundmotion

Vertical earthquake ground motion is not explicitly consideredin the American and Canadian design codes for bridges [1,2]and hence no proposal has been made for vertical earthquakeresponse spectra. It has been discussed in Section 1 that scalingthe horizontal response spectra by 2/3 to get the vertical responsespectra is non-conservative, particularly in the short period range.The recent edition of Eurocode [4] can be considered as a significantimprovement over other design codes. Although vertical responsespectrum is tied with horizontal response spectrum, it recognizesthe differences in frequency contents between horizontal andvertical earthquake ground motion. Soil amplification factorand behavior factor (ductility capacity) for vertical motion areconsidered equal to 1. The V/H peak ground acceleration ratioand spectral ratio are considered as 0.9 and 1.1, respectively, forground type A (rock or other rock like geological formation) whenType 1 response spectrum (earthquake magnitude > 5.5) (Fig. 4)is considered. However, no distinction has been made for viscousdamping against horizontal and vertical motion; although, for thelatter, damping is expected to be significantly lower.In the absence of site specific vertical response spectrum

and until parametrically described vertical response spectra are

F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326 2321

Table 2Values of αi,ψ .

Bridge type ψ R1 R2 R3

Single-span 1.0 3.200 3.200 –Two-span 0.5 4.982 4.597 3.9490

0.6 5.067 4.333 3.98000.7 4.850 4.760 3.85900.8 4.434 4.557 3.96500.9 4.208 4.262 4.02801.0 3.974 4.022 3.9740

Three-span 0.5 5.743 5.278 –0.6 5.601 5.480 –0.7 5.067 5.172 –0.8 4.661 4.803 –0.9 4.427 4.515 –1.0 3.834 4.258 –

Four-span 0.5 5.720 5.026 4.6940.6 5.609 5.491 4.5800.7 5.029 5.142 4.7680.8 4.650 4.782 4.8040.9 4.424 4.576 4.5531.0 3.817 4.318 4.344

proposed in the American and Canadian design codes for bridges, itis recommended to construct vertical response spectrum based onhorizontal response spectrum. Implications for the construction ofvertical response spectrum from horizontal response spectrum arediscussed below.In the AASHTO code [1], the elastic seismic coefficient, Csm, for

the nth mode of vibration shall not exceed 2.5A, where A is theacceleration coefficient. However for soil profile type III (soil profilewith soft to medium-stiff clays and sands, characterized by 30 ftor more of soft to medium-stiff clays with or without interveninglayers of sand or other cohesionless soils) and soil profile typeIV (a profile with soft clays or silts greater than 40 ft) in areaswhere acceleration coefficient is greater than 0.3, Csm shall notexceed 2.0 (Fig. 5). This reduction of Csm is to take into accountthe nonlinear site response effect, as under strong earthquakeshaking soil response is nonlinear [20]. At higher acceleration,the effective shear modulus of soil decreases and hence soildamping increases. The nonlinear behavior of soil causes theamplification factor to be dependent on the type of soil (effectiveshear modulus) and on the intensity of the ground shaking. Theincreased nonlinearity of soft soil response at higher accelerationsreduces the amplification ratio because of the increase in hystereticdamping and the reduction in effective shear moduli [20].However, the vertical ground motion is mainly associated withthe arrival of vertically propagating compressive P-waves, whilsthorizontal ground motion is mainly associated with the secondaryS-waves. There is strong evidence that the vertical ground motionis not strongly influenced by nonlinear site effect in the way thehorizontal groundmotion is [9]. Hence, the reduction of maximumCsm for soil profile types III and IV under strong ground motion canbe ignored for the vertical response spectrum.Similarly to the American code, in Canadian Highway Bridge

Code [2], the elastic seismic coefficient, Csm, for the nth mode ofvibration shall not exceed 2.5AI , where A is the zonal accelerationratio and I is the importance factor (for lifeline bridges I = 3, foremergency-route bridges I = 1.5 and for other bridges I = 1).Also, for soil profile types III and IV in areas where the accelerationcoefficient is greater than 0.3, Csm shall not exceed 2.0. As discussedabove, this reduction in Csm can be ignored for the vertical responsespectrum.Modern seismic design codes recognize that it is uneconomical

to design a bridge to resist large earthquakes elastically. Seismicdesign codes take the advantage of inherent energy dissipationcapacity of the structure and reduce the design horizontal force

1

2

Nor

mal

ized

Res

pons

e S

pect

rum

0

3

1 2

Period (Sec)

0 3

Vertical Elastic Repsonse Spectrum

Horizontal Elastic Response Spectrum for Ground Type A

Fig. 4. Eurocode [4] recommended elastic response spectra for ground type A(Normalized with respect to peak horizontal acceleration, ag ).

1

2C

sm N

orm

aliz

ed D

esig

n C

oeffi

cien

t

0

3

1.0 2.0Period ( second)

0.0 3.0

SOIL PROFILE TYPE IV

SOIL PROFILE TYPE IIISOIL PROFILE TYPE II

SOIL PROFILE TYPE I

NOTE: DOTTED LINES SHOW FORMS OF COEFFICIENTS FOR SOIL TYPE III AND IV WHEN ACCELERATION COEFFICIENT IS LESS THAN 0.3

Fig. 5. AASHTO [1] recommended seismic response coefficient for various soilprofiles (normalized with respect to acceleration coefficient A).

level by dividing it by the R-factor. The energy dissipation (or duc-tility) capacity is achieved by stringent detailing requirements forstructural components that are expected to yield under the ac-tion of a strong earthquake. However, the ductility capacity of thebridge under vertical earthquake is not generally considered [4],and hence the response modification factor is considered hereas 1.0.The importance factor, I , and response modification factor, R,

have been clearly distinguished in Canadian code; whereas inAmerican code, response modification factor combines the socalled ‘‘structural ductility factor’’ with the ‘‘importance factor’’[21]. The intent of the importance factor, I , is to enhance thesafety and performance of important structures. The use of anearthquake importance factor increases the design force level and,in effect, for a given earthquake, lessens the inelastic demand ona structure [22]. The importance factor for a lifeline bridge (I =3.0) may seem very high and results in 3 times the design forcelevel for other (ordinary) bridges. However, this increase is offsetby the use of a high R-factor (up to 5, for ductile RC multiplecolumn bents, ductile steel columns or frames). Since for verticalearthquake groundmotion theR-factor is usually considered as 1.0,it may not be realistic to use importance factor for calculating thesupport reactions of the bridges under vertical earthquake groundmotion. Moreover, Canadian code and American code are expectedto produce similar results. Hence, it is recommended to use theimportance factor (I) as 1.0 for all bridge classes.

2322 F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326

Fig. 6. Example Bridge1: Rigid bridge (a) Elevation (b) Superstructure cross-section.

5. Examples

5.1. Summary of the simplified method for rigid bridges

(i) Identify class of bridge, unit weightw and span length L.(ii) Determine seismic acceleration coefficient (or zonal accel-eration ratio) A [1,2]. Identify the a-value as the maximumresponse spectral acceleration (Fig. 5). For site specific ver-tical acceleration response spectrum, a-value represents theplateau of the response spectrum.

(iii) Determine value of bi from Fig. 2.(iv) Calculate reaction of each support i as: Ri = abiwL.

5.2. Summary of the simplified method for flexible bridges

(i) Identify class of bridge, unit weightw and span length L.(ii) Identify type of soil, determine S and A [1,2], or develop thesite specific vertical acceleration response spectrum.

(iii) Determine value of αi,ψ from Table 2 and compute effectiveperiod for each support:

Teff (i,ψ) =2πα2i,ψ

√wL4

gEI.

(iv) Determine ai from Teff (i,ψ) and design acceleration responsespectrum (Fig. 5).

(v) Determine value of bi from Fig. 2.(vi) Calculate reaction of each support i as: Ri = aibiwL.

5.3. Calculation of support reactions for example bridges

Three bridges are selected to demonstrate the calculation ofsupport reactions using the proposed simplified method and tocompare results with the support reactions obtained from othermethods. Bridge 1 is a two-span slab bridge (Fig. 6) in San Francisco,California (A = 0.6). The bridge has short span length and canbe considered as a stiff bridge. Bridge 2 is a typical river crossingbridge (Fig. 7) with one central pier. The bridge is located inMontreal, Canada (A = 0.2, Site Class I). The bridge has mediumspan length and can be considered as a flexible bridge. Bridge 3is a three-span slab bridge (Fig. 8) in Northern New Jersey (A =0.18). This bridge is chosen to demonstrate the capability of thesimplifiedmethodwhen the bridge has a span ratio (ψ) in betweentwo tabulated values. Support reactions are calculated consideringthe vertical response spectrum derived from horizontal responsespectrum in Refs. [1,2], as discussed in Section 4.

Bridge 1: Rigid bridge

(i) It is a two-span bridge with ψ = 16/20 = 0.8, unit weightw = 160 kN/m and main span length L = 20 m.

(ii) Seismic acceleration coefficient of the bridge site A = 0.6.Hence, a = 2.5× 0.6 = 1.5 [refer Fig. 5].

(iii) From Fig. 2, b1 = 0.33, b2 = 0.63 and b3 = 0.25.(iv) Support reactions are therefore:

R1 = 1.5× 0.33× 160× 20 = 1584 kNR2 = 1.5× 0.63× 160× 20 = 3024 kNR3 = 1.5× 0.25× 160× 20 = 1200 kN.

Bridge 2: Flexible bridge

(i) It is a two-span bridge with ψ = 1.0, L = 60 m. EI =208 000 MNm2 and unit weightw = 174 kN/m.

(ii) The zonal acceleration ratio for Montreal A = 0.2.(iii) Calculation of Teff (i,ψ):

(a) From Table 2, α1 = 3.974, so

Teff (1) =2π3.9742

√174 000× 604

9.81× 208 000× 106= 0.418.

(b) From Table 2, α2 = 4.022, so

Teff (2) =2π4.0222

√174 000× 604

9.81× 208 000× 106= 0.408.

(iv) Determination of ai:(a) at T1 = 0.418, a = 0.429 (Ref. [2])(b) at T2 = 0.408, a = 0.436 (Ref. [2])

(v) From Fig. 2, b1 = 0.31, b2 = 0.89, b3 = 0.31.(vi) Support reactions are therefore, R1 = 0.429 × 0.31 × 174 ×

60 = 1388 kN; R2 = 0.436× 0.89× 174× 60 = 4051 kN

by symmetry, R3 = R1 = 1388 kN.

Bridge 3: Three-span slab bridgeConsidering the bridge as a rigid bridge, the support reactions

are calculated as follows:

(1) It is a three-span bridge with ψ = 23.1/30 = 0.77.Calculations are performed for ψ = 0.7 and ψ = 0.8. It hasbeen found reasonable, based on several studies, to interpolatevalues when actual ψ is not tabled. The unit weight of thebridge is 148 kN/m and L = 30 m.

(2) Seismic acceleration coefficient (A) of the bridge site inNorthern New Jersey is 0.18. Hence, from Eq. (15), a = 2.5 ×0.18 = 0.45.

F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326 2323

Fig. 7. Example Bridge 2: Flexible bridge (a) Elevation (b) Superstructure cross-section.

Fig. 8. Example Bridge 3: Three-span slab bridge elevation.

(3) From Fig. 3, b1 = 0.26 and b2 = 0.58+(0.69−0.58)×(0.77−0.7)/0.1 = 0.657.

(4) Support reaction is therefore: R1 = 0.45× 0.26× 148× 30 =520 kN; R2 = 0.45× 0.657× 148× 30 = 1313 kN.

Modal analyses using computer program SAP 2000 have beencarried out to investigate the accuracy of the proposed methodfor the three selected bridges. Mode shapes for the first fourdynamic modes (vertical) have been shown in Tables 3–5. ForBridge 1, the dominant vertical vibrationmode is the secondmodewith modal mass participation factor 0.65 (Table 3). The modalmass participation factor for the first mode is significantly lowerthan the first mode. Similarly to Bridge 1, the dominant verticalvibration mode for Bridge 2 is the second mode with modal massparticipation factor 0.81 (Table 4). The modal mass participationfor the first mode has been observed to be negligible. For Bridge3, the dominant vertical vibration mode is the third mode withmodal mass participation factor 0.79 (Table 5). Similarly to Bridge2, modal mass participation factor for the first mode is negligible.Support reactions obtained from modal analyses are in very

good agreement with the results obtained from simplified method(Table 6), when the supports of the bridge are considered rigid. Itis important to note that the developed simplifiedmethod is basedon the assumption that the supports are rigid. However, if the verti-cal stiffness of the supports is considered, it has been observed thatsupport reactions obtained from SAP 2000 are only slightly higherthan those obtained from simplified method, except for the midsupport of Example Bridge 1. It may be conservative to introduce afactor of 1.1 to take into account the flexibility of supports.

It is often argued that the single mode spectral method(Rayleigh method) as prescribed in the design codes for horizontalground motion may be used to calculate the effect of verticalground motion. However, the accuracy of the Rayleigh method issignificantly dependent on the assumed deformed shape of thebridge under uniform vertical load. Hence, calculations of supportreactions using Rayleigh method can be readily investigated bycomparing the dominant mode shape with the deformed shape ofthe bridge under uniform vertical load.It is evident from the comparison of the deformed shape

of the bridge under uniform vertical load (Tables 3–5) thatRayleigh method fails to assume the proper shape function. Thesupport reactions obtained from Rayleighmethod are significantlydifferent from modal analyses results (Table 6). Hence, it isproposed not to rely on Rayleigh method when support reactionsunder vertical earthquake ground motion are to be calculated.Notional support reactions under vertical earthquake ground

motion prescribed in the design code [2] by applying a factorof 0.25 to the dead load action are found to be much lowerthan the values obtained from simplified method or modalanalyses (Table 6). Hence, a factor of 0.25 or less would certainlyunderestimate the support reactions under vertical earthquakeground motion. The underestimation would be even worse forhigher shaking levels.It can be observed that the proposed simplified method can be

used to calculate the support reactions when the bridge has a spanratio (ψ) in between two tabulated values (Table 6 for Bridge 3).It may be argued that the response spectra used to verify the

methodology may not be representative of typical vertical ground

2324 F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326

Table 3Mode shapes of Example Bridge 1: Rigid bridge.

Mode Mode shape Modal mass participation factor(vertical direction) (%)

1 16

2 65

3 2

4 0

Deformed shape under uniformvertical load

Table 4Mode shapes of Example Bridge2: Flexible bridge.

Mode Mode shape Modal mass participation factor(vertical direction) (%)

1 0

2 81

3 0

4 1

Deformed shape under uniformvertical load

motion. Calculations have been repeated based on the verticalresponse spectral shape suggested in Eurocode [4] (Fig. 4). Peakground accelerations of California, Montreal and New Jersey areconsidered as 0.6g , 0.2g and 0.18g , respectively. The results havebeen presented in Table 6. Similar observations have been noticedimplying that the developedmethodology is not dependent on theshape of the response spectra.

6. Conclusions and recommendations

A theoretical approach to calculate support reactions of bridgesunder vertical earthquake ground motion has been presented.Based on this theoretical approach, a simplified method hasbeen developed that enables the calculation support reactionsefficiently with less computational efforts. The method can beeasily adopted in the seismic design of highway bridges.Using the developed simplified method, the dominant vertical

modal period of the bridge can be calculated, which can be used tocheck finite element analysis result or to make sure that the finiteelement model does not exclude the important vibration modes inthe vertical direction. This is certainly important since it is often

difficult to get sufficient contributing vertical modes in a three-dimensional model.Incorporation of a percentage of dead load action for the vertical

earthquake ground motion may provide highly non-conservativeresults even for moderate seismic demand. Hence, it would not beprudent for the design codes to use a dead loadmultiplier for takinginto account the effect of vertical earthquake ground motion.Based on the results obtained in this paper, single mode

spectral method (Rayleigh method) may not be applicable for thecalculation of support reactions under vertical earthquake groundmotion.In the development of the simplifiedmethod, stiffness contribu-

tions from bearings, piers and foundations have not been consid-ered. Modal analysis by computer program SAP 2000 with similarassumptions provides very similar results. It has been shown thatevenwhen foundation flexibility is incorporated, support reactionscalculated by the proposed simplified method are approximatelywithin 10% of a full modal analysis using SAP 2000. To be conser-vative a factor of 1.1 can be appliedwith the results obtained by thesimplified method developed herein when supports are assumedflexible.

F. Legeron, M. Neaz Sheikh / Engineering Structures 31 (2009) 2317–2326 2325

Table 5Mode shapes of Example Bridge 3: Three-span slab bridge.

Mode Mode shape Modal mass participation factor(vertical direction) (%)

1 1

2 0

3 80

4 0

Deformed shape under uniformvertical load

Table 6Support reactions of example bridges using vertical response spectrum derived from the horizontal response spectrum in Refs. [1,2] and the vertical response spectrumrecommended in Eurocode [4].

Bridges Analysis method Vertical response spectrum derived fromhorizontal response spectrum in Refs. [1,2]

Vertical response spectrumrecommended in Eurocode [4]

R1 (kN) R2 (kN) R3 (kN) R1 (kN) R2 (kN) R3 (kN)

Example Bridge 1 Simplified method 1584 3024 1200 1711 3266 1296Modal analysis-rigid support 1543 3089 1165 1657 3301 1242Modal analysis-flexible support (vertical stiffness of endsupports= 2650 MN/m and mid support= 4210 MN/m)

1762 3712 1294 1890 3993 1357

0.25× dead load reaction 215 908 316 215 908 316Rayleigh method 116 2995 1846 125 3235 1995

Example Bridge 2 Simplified method 1388 4051 1388 628 1849 628Modal analysis-rigid support 1318 3919 1318 631 1899 631Modal analysis-flexible support (vertical stiffness of endsupports= 2650 MN/m and mid support= 4210 MN/m)

1350 4068 1350 683 2107 683

0.25× dead load reaction 980 3260 980 980 3260 980Rayleigh method 1147 3799 1147 502 1689 502

Example Bridge 3 Simplified method 520 1313 561 1417Modal analysis-rigid support 512 1337 546 1432Modal analysis-flexible support (vertical stiffness of endsupports= 2650 MN/m and mid support= 4210 MN/m)

539 1449 543 1471

0.25× dead load reaction 312 1097 366 2147Rayleigh method 525 3077 525 3077

Acknowledgments

The research work described in this paper commenced sincethe first author worked at SETRA-Large Bridge Division in France.Invaluable contributions from several individuals of SETRA in-cluding P. Corfdir, M. Kahan, R. Tardy, T. Kretz, E. Bouchon andA. Chabert, are gratefully acknowledged. Invaluable contributionby undergraduate student Mathieu Pacocha has been gratefullyacknowledged. Financial support from Natural Science and En-gineering Research Council (NSERC) of Canada is also gratefullyacknowledged.

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