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1

Simplified design of bridges for multiple-support earthquake excitation

Savvas P. Papadopoulos a and Anastasios G. Sextos b

a Civil Engineer, Ph.D. Candidate, Division of Structural Engineering, Department of Civil Engineering,

1

Aristotle University Thessaloniki, Greece; e-mail: savvaspp@civil.auth.gr

2

b Professor, Department of Civil Engineering, University of Bristol, UK (a.sextos@bristol.ac.uk)

3

(*Corresponding Author)

4

ABSTRACT

5

This paper presents a novel, bridge-dependent approach for quantifying the increase of design

6

quantities due to spatially variable earthquake ground motion (SVEGM). Contrary to the existing

7

methods for multiple support bridge excitation analysis that are either too complicated to be

8

applied by most practitioners or oversimplied (e.g. Eurocode 8, Annex D provisions), this method

9

aims to strike a balance between simplicity, accuracy and computational efficiency. The method

10

deliberately avoids generating support-dependent, acceleration or displacement, asynchronous

11

inputs for the prediction of bridge response. The reasons behind this decision are twofold: (a)

12

first, the uncertainty associated with the generation of asynchronous motion scenarios, as well as

13

the exact soil properties, stratification and topography is high while, (b) the response of a bridge is

14

particularly sensitive to the above due to the large number of natural modes involved. It is therefore

15

prohibitive to address SVEGM effects deterministically in the framework of a design code. Instead,

16

this new method is based on two important and well-documented observations: (a) that SVEGM

17

is typically globally beneficial but locally detrimental [1], and (b) that the local seismic demand increase

18

is very closely correlated with the excitation of higher modes, which are not normally activated in the

19

case of uniform ground motion [2,3]. Along these lines, a set of static analyses are specified herein

20

to complement the standard, code-based response spectrum analysis. These static analyses apply

21

spatially distributed lateral forces, whose patterns match the shape of potentially excited anti-

22

symmetric modes. The amplitude of those forces is derived as a function of the expected

23

amplification of these modes according to the process initially proposed by Price et al. [4]. Two

24

real bridges with different structural configurations are used as a test-bed to demonstrate the

25

effectiveness of the new method. Comparison of the results with those obtained through rigorous

26

response history analysis using partially correlated, spatially variable, spectrum-compatible input

27

motions [5] shows that, the simplified method presented herein provides a reasonably accurate

28

estimation of the SVEGM impact on the response of the bridges examined at a highly reduced

29

computational cost. This is essentially an elastic method that is found to be simple, yet precise

30

enough to consist an attractive alternative for the design and assessment of long and/or important

31

bridge structures in earthquake-prone regions.

32

Keywords: bridges, multiple-support excitation, spatial variability, anti-symmetric modes, seismic

33

codes

34

35

2

1. INTRODUCTION

1

Bridges are seismically vulnerable components of a transportation network and they can cause

2

severe adverse socioeconomic consequences in case of failure [6]. As they often cross irregular

3

topographic profiles at long distances, they are much more susceptible to the effects of multiple-

4

support excitation in comparison to other structures, such as buildings for instace. The latter relates

5

to the fact that their support points are discrete and they can be separated by several meters apart,

6

hence, seismic motion accross successive piers may vary significantly in terms of arrival time,

7

frequency content and amplitude. This phenomenon introduces a ground motion-related

8

uncertainty that is additional to the standard record-to-record variability particularly for long

9

structures and/or abruptly changing soil profiles [7]. Spatial variability of earthquake ground

10

motion (SVEGM) and its effect on seismic performance of bridges has long been studied [8], while

11

field evidence from previous earthquakes (e.g. Loma Prieta, Kobe) has highlighted its potentially

12

detrimental effects [6,9,10]. Careful processing of data from dense seismographical arrays clearly

13

demonstrates the main sources of asynchronous motion at the base of bridge piers, namely, [11]:

14

(a) the effect of wave passage, that is the finite time required for seismic waves to reach and excite

15

successive support points, (b) loss of coherency (i.e., loss of statistical correlation) of the

16

propagating seismic waves, (c) local site effects, (d) geometrical attenuation of the seismic waves

17

with diastance, and (e) kinematic soil-structure interaction that leads to local (i.e., pier-dependent)

18

filtering of higher frequencies.

19

It is now common belief that due to SVEGM being strongly case-dependent, deterministic

20

approaches are not adequate to capture the dispersion of structural response attributed to the

21

above phenomena, hence, the problem needs to be studied in a probabilistic manner. Different

22

methods have already been developed for that purpose, each one exhibiting its own shortcomings

23

and limitations. Random vibration analysis (RVA) has been extensively used to quantify the

24

sensitivity of different types of bridges, such as highway [12,13], suspension [14,15] and cable-

25

stayed [14–17] bridges to multi-support excitation. RVA, although consistent with a statistical

26

characterization of the response [18], which is key for performance-based design [19], is

27

unfortunately too complex to be used for most practical purposes. Fundamental principles of

28

random vibration theory have also been exploited as an extension of the response spectrum

29

method for the case of non-synchronous input motion [4,11,18,20–24]. In this context, the Multi-

30

Support Response Spectrum method, introduced by Der Kiureghian & Neuenhofer [11] and

31

extended later by Konakli & Der Kiureghian [18], appears to be the most accurate, with the ground

32

motion characterized by the response spectrum and the response quantities calculated as the mean

33

3

of their peak values. A limitation of both RVA and the response spectrum-based methods,

1

however, is that they are inherently solving linear or linearized problems, hence they cannot be

2

easily used for assessing the seismic capacity of existing bridges.

3

On the other hand, time history analysis (THA) employing partially correlated synthetic

4

accelerograms seems to be a more straightforward option for estimating the structural response in

5

a Monte Carlo framework. In this context, ground motions can be simulated through a variety of

6

existing techniques [25–39], while different approaches have been proposed in the literature to

7

satisfy the seismic codes' requirement for compatibility between the simulated ground motion

8

suites and the target response spectrum [5,26,32,39–43]. Since THA can be used for all degrees

9

and sources of nonlinearity (i.e., both material and geometric), it has been used in several studies

10

for the case of different bridge types [1,3,7,9,10,44–57] at the price of course of computational and

11

post-processing time of the large number of motions that need to be generated, converge to the

12

target coherency and frequency content and be applied at different support points.

13

An additional dimension of this problem is that despite the extensive research discussed above,

14

currenty, there is no clear trend as per the (beneficial or critical) impact of non-uniform input

15

motion on bridges. If the authors need to make a general statement this would be that SVEGM

16

has an overall favorable effect on average, which can be also locally detrimental as significant response

17

amplifications can also be triggered on specific bridge locations and components (i.e., stoppers,

18

specific piers, cables etc). In fact, the influence of SVEGM on bridge response is a complex

19

problem, depending on the engineering demand parameter (EDP) of interest, its location on the

20

structure, the dynamic characteristics of the bridge in question and the assumptions made regarding

21

the seismic input scenario in terms of ground motion correlation, soil conditions etc. An inherent

22

consequence of this complexity is that it is impossible to deterministically predict response

23

quantities due to asynchronous motion based on scaling [10] or combination [1] of uniform ones.

24

A second important issue related to the effect of SVEGM is the mechanism associated with the

25

dynamic response of soil-bridge systems under multiple-support excitation. Even though it has

26

been long recognized that the total displacement of a multiply-supported MDOF system is the

27

sum of a pseudo-static and a dynamic component [8], it is the latter that tends to trigger local

28

seismic demand increase due to the excitation of higher anti-symmetric modes. This fact was

29

observed by Harichandran & Wang who employed RVA to study the response of single [58] and

30

two span [59] continuous beams under SVEGM. Zerva [57,58] reached a similar conclusion for

31

the case of N-support continuous beams excited by partially correlated motions, with the second

32

study additionally considering the wave passage effect. Contribution of anti-symmetric modes to

33

4

the total response has also been observed for asynchronously excited suspension [14,15,60] and

1

arch [15] bridges. Studies based on THA further revealed the de-amplification of symmetric modes,

2

particularly of the first transverse one in most cases, and the respective amplification of the anti-

3

symmetric ones in the case of straight [1,9,10,47,54,61–63], curved [10,44] and cable-stayed

4

[3,50,64] bridges. Recently, Sextos et al. [3] presented a study for the case of the Evripos cable-

5

stayed bridge using real, free-field and superstructure recordings obtained during the (Ms=5.9,

6

1999) Athens earthquake providing measured evidence that verified the excitation of higher,

7

primarily antisymmetric, modes of vibration. This observation is also in line with experimental

8

studies conducted for the case of straight [65,66] and curved [67] bridges. In this context,

9

Papadopoulos & Sextos [2] quantified the excitation of anti-symmetric modes and correlated them

10

with bridge response quantities. Common ground in the aforementioned studies is that the

11

amplification and de-amplification of the bridge response quantities is in very good agreement with

12

the excitation of higher anti-symmetric modes and the reduced vibration at the predominant

13

structural modes, respectively.

14

Despite the aforementioned progress, most seismic codes worldwide do not yet address the

15

SVEGM through a solid approach for the generation and application of spatially variable ground

16

motion suites. Instead, indirect measures, such as larger seating deck lengths and simplified

17

methods are employed. Currently, only two seismic codes (Eurocode 8 – Part 2 for bridges [53]

18

and the New Italian Seismic Code [68]) explicitly deal with the SVEGM; their provisions, aim at

19

capturing solely the increased bridge demand due to the pseudo-static response component,

20

ignoring the excitation impact of higher anti-symmetric modes. In addition, the proedure presented

21

in the current version of Eurocode 8 [53], fails to provide anything but a very minor effect on the

22

predicted design quantities while, by its own static nature, cannot be applied for bridges which are

23

insensitive to statically imposed displacements, such as those designed with seismical isolation [7].

24

It can thus be concluded that modern seismic codes in the U.S., Europe and Asia do not provide

25

as yet a comprehensive framework for the consideration of SVEGM in the seismic design and

26

assessment of bridges due to the lack of a simple, theoretically sound and code-oriented approach,

27

which will be easily applicable and thus appealing to the civil engineering community.

28

In this context, the objective of this paper is to present a novel, bridge-dependent, simplified

29

approach for designing bridge structures for multi-support earthquake excitation. The method,

30

which refers to the lateral response of the bridge, aims to offer a different view for solving this

31

complex problem as summarized below:

32

5

(a) it does not generate suites of spatially variable earthquake ground motions to predict the

1

structural response, on the grounds that this involves complex simulation procedures and

2

uncertain input that lead to bridge responses that are very sensitive to the assumptions

3

made

4

(b) it accepts a-priori that: if SVEGM effects are significant (e.g. for the case of long bridges

5

or bridges crossing abruptly changing soil profiles), then it is only the bridge piers that are

6

associated with the excitation of anti-symmetric higher modes that are detrimentally

7

affected at a local level.

8

(c) with that in mind it aims to directly quantify the local seismic demand amplification and

9

apply it as an additional safety margin to critical piers only without actually running a SVEGM-

10

based THA.

11

The rational behind this method and the mathematical formulation is presented in the following

12

along with its application for the case of two real bridges for demonstrating its applicability and

13

accuracy against a rigorous asynchronous excitation THA.

14

2. EQUATIONS OF MOTION & CODE PROVISIONS

15

2.1 Equations of motion

16

The assumption of uniform excitation at the support points involving M-degrees of freedom of an

17

N-degrees-of-freedom (N-DOF) system (Fig. 1) is no longer valid in the case of bridge structures

18

of significant length and for those crossing different soil profiles or irregular topographies. Table

19

1 summarises the differential equation which governs the response of a system under uniform and

20

non-uniform ground motion (Eq.1-8). In these equations Μ, C, and K are the respective [N×N]

21

mass, damping and stiffness matrices of the N unconstrained DOFs, Μg, Cg, and Kg are the

22

respective [M×M] mass, damping and stiffness matrices associated with the M-DOF at the

23

supports, Μc, Cc, and Kc are the respective [N×M] coupling mass, damping and stiffness matrices

24

between the N unconstrained and the M constrained DOF and ut and ug are the {N×1} and

25

{M×1} vectors of displacements of the unconstrained and the constrained DOF, respectively. In

26

both cases of uniform and non-uniform excitation, the total displacements are decomposed into

27

their dynamic and pseudo-static components (Eq.2). In the first case of uniform motion, r is the

28

{N×1} influence vector, which represents the rigid body displacements of the masses related to

29

the active direction of the support motion, while in the second case, R is the [N×M] influence

30

matrix, where each column {rk} represents the static displacements of the unconstrained DOF

31

6

when the kth support experiences unit displacement while all other supports are fixed. Substituting

1

Eq.2 in Eq.1, considering a lumped mass model (Mc=0) and ignoring the damping terms of the

2

effective force vector in the case of non-uniform excitation as negligible, the equations of motion

3

are significantly simplified (Eq.3). In terms of the latter, the damping term equals zero if the

4

damping matrices are proportional to the stiffness matrix, otherwise it is usually small enough, in

5

relation to the inertia term, to be ignored [69]. In addition, expanding the displacements u in terms

6

of modal contribution (Eq.4), with qi being the modal coordinates, substituting them into Eq.3 and

7

taking advantage of the modes' orthogonality, the N decoupled equations of motion are derived

8

(Eq.5). In these equations Γi is the modal participation factor of mode i when the structure is

9

uniformly excited and and Γi,k the modal participation factor of the same mode related to the

10

excitation of the kth support under multi-support excitation. It is important to note that, in the case

11

of a structure subjected to uniform input motion, the modal participation factor of mode i is

12

defined as: .

13

14

Figure 1. Multi-degree of freedom models under uniform (on the left) and non-uniform (on the right)

15

excitation.

16

Table 1. Comparative presentation of the equations of motion for uniform & non-uniform excitation.

17

Uniform excitation

Multi-support excitation

(1)

(2)

(3)

(4)

,

1=

G=G

åM

iik

k

=

t

Mu + Cu + Ku 0

!! !

éùéùìüéùìüìü

ìü +

íý íý íýíý

êúêúêú

îþ

ëûëûîþëûîþîþ

t

g

cc ct

t

TTT

cg cgg cgg g

MM CC u KK u 0

u+=

MM CCu KKu P

u

!

!!

!

!!

=+ =+

ts

uuuur

g

u

ìü

ìü ìü ì ü

ìü ìü ìü

ïï

++ +

íýíýíýíýí ýíýí ý

ïï

îþ îþ îþ

îþ îþ î þ

îþ

-1

ts g

cg

gg g

g

uu Ru

-K K u

uu u

== =

uu u

u

00 0

g

u++ =-Mu Cu Ku Mr

!! ! !!

1

M

gk

k

u

=

-å

Mu + Cu + Ku = Mr

!! ! !!

k

1

N

i

i

q

=

==

å

uΦq φi

1

N

i

i

q

=

==

å

uΦq φi

7

(5)

(6)

(7)

(8)

u; ut; us; ug; ug; ug,k; qi; qi,k; Di; Di,k and their derivatives are time dependent functions (e.g. u(t))

1

Eq.6 is the solution of the ith (i=1,…N), decoupled equation of motion, where Di(t) is the response

2

of a single DOF oscillator with the dynamic characteristics of mode i, subjected to the excitation

3

of the kth support. The dynamic displacements of the system are shown in Eq.7 and the total

4

displacements are given by Eq.8. It can be seen therefore that the dynamic response of an extended

5

structure is considerably different for multi-support and identical input motion excitation. In the

6

first case, the pseudo-static displacements (first term of Eq.8) do not produce any elastic forces as

7

they represent rigid body motion, while the opposite is valid for the respective term under non-

8

uniform motion. However, the difference is not limited to the additional pseudo-static internal

9

forces, but extends to the dynamic component as well, through the different modal participation

10

factors associated with mode i and excitation of the kth support DOF. It is worth noting again that

11

the latter, despite its direct impact on the dynamic component of the response, is ignored in the

12

simplified provisions of Eurocode 8 as discussed below.

13

2.2 EC8 provisions for SVEGM

14

EC8-Part 2 for bridges [53] provides a simplified framework in its informative Annex D for

15

considering SVEGM in the form of additional pseudo-static internal forces. Two characteristic

16

relative displacement patterns between the supports are considered therein: (a) piers subjected to

17

ground displacements of the same sign but varying amplitude (SET A):

18

(9)

19

and (b) successive piers displaced in opposite directions (SET B):

20

2

2++=-G

G=

T

T

φMr

φMφ

!! ! !!

iiiiii ig

i

qqqu

wz w

i

ii

{ }

2

,,

1

,

2

=

++=-G

G=

å

T

T

φMr

φ Mφ

!! ! !!

M

iiiiii ik gk

k

ik

qqq u

wz w

ik

ii

iii

qD=G

,,,

11

MM

iik ikik

kk

qq D

==

=G=

åå

1

N

ii

i

D

=

== G

å

uΦq φi

,,

1

1

M

N

ik ik

k

i

D

=

=

== G

å

å

uΦq φi

1=

=+ G

å

t

ur φ

N

gii

i

uD

i

{ }

,,,

1

11

M

MN

gk ik ik

k

ki

uD

=

==

=+G

å

åå

t

ur φ

ki

22

i

ri g g

g

L

dd d

L

=£

8

(10)

1

where dg is the design ground displacement corresponding to the soil type at support i, Li is the

2

distance from pier i to the reference point (usually the abutment), Lg is the distance beyond which

3

the motion is considered uncorrelated and Lav,i is the average distance between Lavi,i-1 and Lavi,i+1. The

4

stresses imposed by the above two sets of pseudo-static forces (Ei,SetA, EiSetB) are derived through

5

static analysis. The maximum action effect arising from the two distinct static cases is then

6

superimposed with the outcome of a typical response spectrum analysis (Ei,in) by means of the

7

square root of the sum of squares (SRSS) combination rule:

8

(11)

9

Comparative studies on the effectiveness of this simplified approach with more sophisticated ones

10

have highlighted important logical and theoretical issues that may, under certain circumstances,

11

lead to highly unconservative design [7]. In addition, being effectively static in nature, the above

12

force patterns cannot be applied on bridges which are insensitive to statically imposed

13

displacements, such as for instance seismically isolated bridges.

14

An improved process based on the EC8 provisions was proposed by Sextos & Kappos [7] while a

15

alternative to EC8-Part 2 Annex D procedure was developed by Nuti & Vanzi [13,70]. Recently

16

Falamarz-Sheikhabadi & Zerva [71] introduced a deterministic approach for the derivation of

17

simplified displacement loading patterns that can be used in lieu of those proposed by EC8 and

18

irrespectively of the source-to-site distance. These patterns incorporate the propagation

19

characteristics of seismic waves, i.e. the loss of coherency and the wave passage effect, while also

20

taking into account the contribution of the short period waves on the out-of-phase response of

21

adjacent piers. However, similar to the previous methods, its use is limited to bridges which are

22

sensitive to statically imposed displacements and respond in the linear range.

23

3. SIMPLIFIED METHOD TO DESIGN BRIDGES ACCOUNTING FOR

24

SVEGM EFFECTS

25

A novel method is proposed herein to capture the effect of SVEGM. Instead of generating spatially

26

variable acceleration time histories for each bridge support or imposing displacement patterns at

27

the base of each pier, a set of spatially distributed lateral force Fi (i=1, 2, ...) profiles, whose patterns

28

match the shapes of anti-symmetric modes i (i=1, 2, …), is applied on the structure. The respective

29

set of static analyses is subsequently performed and the responses derived are combined with the

30

,

12

2

av i

irg

g

L

dd

L

b

=±

22

,, ,

[max(E , E )]

sd i SetA i SetB i in

E= +E

9

inertial response due to uniform excitation; the latter is calculated through standard methods that

1

are prescribed in the codes, i.e., either by means of the response spectrum or time history analysis.

2

In this context, the lateral force pattern for each mode i is proportional to the product of its mode

3

shape

φi

,, its circular frequency squared and the mass assigned to the nodes M:

4

(12)

5

where Γi is the modal participation factor of mode i, Di is the spectral displacement of mode i and

6

is a scale factor accounting for the amplified contribution of anti-symmetric mode i due to the

7

SVEGM.

8

In the following, the proposed method is analytically presented and discussed in a stepwise manner;

9

a total set of eleven steps is unfolded. More specifically, Step 1 defines the earthquake scenario

10

and wave propagation characteristics assumed (power spectum density, coherency model, Vapp).

11

Step 2 determines the dynamic properties of the bridge through modal analysis. In Step 3, a

12

number of static analyses, equal to the number of supports along the examined direction, is

13

performed and the influence matrix R=-K-1Kc ([NxM]) is constructed. Step 4 computes the modal

14

participation factors Γi,k (Eq.5b) for each mode i. With respect to the examined scenario (Step 1),

15

the non-dimensional spatial variability parameter Ψ(ω) is calculated in Step 5. Step 6 then estimates

16

the generalized participation factors Βi (i=1,…N) for each mode and their upper bound Bimax; the

17

ratios |Bi(f)|/Bimax, are used to estimate the potentially amplified contribution of anti-symmetric

18

modes due to asynchronous earthquake motion which is key to predict the unfavorable effect of

19

spatial variability on individual Engineering Demand Parameters and design quantities.

20

These ratios are then compared with the respective ones under uniform excitation|Γi|/Bimax and

21

the frequency-dependent scale factors SFi(ω) of response due to asynchronous excitation are

22

derived in Step 7. Next, Step 8 estimates the mean values for each mode across the pre-

23

defined excitation frequency range to get an average value of response amplification. In Step 9, a

24

set of static analyses is performed, only for those modes for which >1 (i.e., the effect of spatial

25

variability is detrimental) with loads calculated through Eq.12. Conventional dynamic analysis

26

under uniform excitation is performed in Step 10. Finally, Step 11 combines the response quantities

27

derived from Steps 9 and 10 by means of the SRSS rule. The details of the successive calculation

28

steps are given below.

29

2

i

w

( )

( )

2

1=-×G×× i

FMφ

ii iii

SF D

w

i

SF

i

SF

i

SF

10

3.1 Step 1: Ground motion intensity, coherency and propagation velocity

1

Similarly to all existing methods for multi-support excitation of bridges, a power spectrum density

2

(PSD) and a coherency model need to be defined to characterise the random field of the generated

3

seismic ground motions. In the present method, the sole purpose of the PSD lies in the selection

4

of an excitation frequency range for which the desired (target) level of power is exceeded. To this

5

end, unless a specific PSD corresponding to the region of interest is provided or a method for the

6

conversion of a response spectrum to a PSD is followed, the Clough & Penzien [72] spectrum can

7

readily be used:

8

(13)

9

where So is a constant determining the intensity of acceleration at the dedrock level, ωg and ζg are

10

the characteristic frequency and damping ratio of the ground respectively, and ωf and ζf are

11

additional filtering parameters. The Clough & Penzien [72] spectra for soft, firm and medium soil

12

using the parameters reported by Der Kiureghian & Neuenhofer [11] are illustrated in Fig.2. For

13

sites with soil conditions that greatly vary along a bridge, the mean of the individual PSD spectra

14

can be used.

15

16

Figure 2. Left: The Clough & Penzien spectra for soft, firm and medium soil using the parameters reported

17

by Der Kiureghian & Neuenhofer (in the embedded table). Right: The Luco & Wong coherency model for

18

different values of λx (Vs=1000m/s).

19

The correlation pattern of the spatially variable generated ground motions also needs to be defined.

20

The semi-empirical formulation proposed by Luco & Wong (1986) [73] can be used as the

21

coherency model:

22

( )

( )

()

( )

()

( )

( )

( )

()

( )

24

2

22

222 2

22

14

141 4

+

=

-+-+

gg f

o

ggg f ff

SS

zww ww

w

ww z ww ww z ww

11

(14)

1

where µ is a measure of the relative variation of the elastic properties in the medium, H is the

2

distance in the medium traveled by the shear waves, and ro is the scale length of the random

3

inhomogenities along the path. The ratio of the dimensionless factor λx to the shear wave velocity

4

VS is the drop coherency parameter controling the exponential decay of the function; the higher

5

the ratio the more significant the loss of coherency. According to Luco & Wong [73], a reasonable

6

value for the ratio varies between (2-3)x10-4m-1s, while the dimensionless parameter λx typically

7

varies in the range 0.02-0.5 [74]. Fig.2 illustrates the Luco & Wong [73] coherency model for

8

different values of λx. The wave passage effect is taken into account through the apparent wave

9

propagation velocity Vapp which typically varies between 1000-3000m/s (for more information see

10

[8]).

11

3.2 Step 2: Modal analysis of the bridge

12

A finite element (FE) model of the bridge is developed as usual. Any structural analysis code can

13

be used as long as the mass M ([NxN]) and stiffness matrices K ([NxN]) can be exported. A modal

14

analysis follows, in order to determine the natural periods of the bridge Ti, i=1, …, N and its

15

respective modes

φi

({Nx1}). Matrices M, K and Φ ([ΝxN]) are then used to calculate the modal

16

participation factors Γi,k (according to eq.15) in Step 4 and the generalized participation factors Bi

17

(eq.25 & eq.26) in Step 5 by means of any mathematical tool (e.g. Matlab [75]).

18

3.3 Step 3: Influence matrix R (M static analyses)

19

Having developed the finite element model of the bridge in Step 2, a number of static analyses are

20

performed to construct the influence matrix ([NxM]). The number of analyses

21

required equals the number of bridge supports (= M) along the examined direction. The way this

22

is derived can be easily understood on the basis of matrix R interpretation: each column {rk} of

23

matrix R represents the static displacements of the structure’s unconstrained DOF when its kth

24

support experiences unit displacement while the rest ones remain fixed.

25

3.4 Step 4: Modal participation factors

Γi,k

26

In this step, the modal participation factors for each mode i (i=1,…N) associated with the kth

27

support excitation (k=1,…M) are calculated as:

28

( ) ( ) ( )

20.5

,exp

éù

=-=H

ëû

xS x o

Vwhere r

gxw lwx l µ

=-1

c

R -K K

12

(15)

1

where eigenmodes

φi

{Nx1} and mass matrix M [NxN] are determined in Step 2, and {rk} is the

2

kth column of the influence matrix R computed in Step 3. Equation can be used to

3

verify whether Γi,k has been accurately calculated.

4

3.5 Step 5: Non-dimensional spatial variability parameter Ψ(ω)

5

In order to quantify the amplification of the anti-symmetric modes due to SVEGM, a process

6

similar to the one used by Price & Eberhard [4] is followed. Ground motions at the supports U(ω)

7

(forming an {Mx1} array) are described in the frequency domain and in relation to the “known”

8

motion at the reference point Uο(ω), which could be taken at one of the abutments:

9

(16)

10

where Ψ(ω) ({Mx1} array) is a non-dimensional spatial variability parameter. This non-

11

dimentional parameter effectively expresses the variation of the generated ground motions in the

12

frequency domain with respect to the reference (‘known’) motion at one of the abutments and is

13

derived according to the following procedure.

14

Since the Fourier transformation is used for the analysis of the amplitude and the phase of strong

15

ground motion, a sinusoidal wave function can be assumed for each excitation frequency ωo [71]:

16

(17)

17

where A(ωo) is the amplitude that corresponds to the excitation frequency ωo, which is assumed to

18

remain constant between the piers, x is the distance between each examined support and the

19

reference point, φο is the initial phase, and Rθ is given by:

20

(18)

21

{ }

,

Γ=

T

T

φMr

φMφ

ik

ik

ii

,

1=

G=G

åM

iik

k

( )

( )

( )

( ) ( )

( )

( )

( )

11

U

UU

UΨ

MM

Y

éù éù

êú êú

== =

êú êú

êú êú

ëû ëû

UωΨω!!

oo

ww

ww

ww

( ) ( ) ( )

,sin=A+ +

ooo

uxt t Rx

qo

wwwj

2

22

21

=+

x

Sapp

RVV

q

l

13

where λx is the dimensionless factor selected in Step 1, controling the exponential decay of the

1

coherency function, VS is the shear wave velocity in the soil medium, and Vapp is the apparent wave

2

propagation velocity (reasonable values 1000-3000m/s, but for more information see [8]).

3

Expressing Eq.17 in the frequency domain, the motion at each pier due to the excitation frequency

4

ωo is given by:

5

(19)

6

which, in the case of the reference point, takes the form:

7

(20)

8

Substituting Eq.19 and Eq.20 into Eq.16, each element Ψx(ω) (x=1,…,M) of Ψ(ω) results in:

9

(21)

10

In the above equation, the non-dimensional spatial variability parameter Ψx(ω) depends on the

11

assumed ground motion model of Step 1. In this context, calculation of Ψx(ω) is made as follows:

12

(a) In the conventional case of fully coherent ground motions along the bridge supports where

13

wave passage is ingored (i.e., uniform excitation), λx=0, Vapp→∞ hence, Ψx(ω)=1.

14

(b) In the idealized case where wave passage effect is indeed accounted for but ground motions

15

are fully correlated, the waveform travels with finite apparent propagation velocity Vapp, the

16

coherency drop parameter is λx=0 and Ψx(ω) is defined as:

17

(22)

18

(c) In the case that both wave passage and incoherency are accounted for, Ψx(ω) depends on the

19

assumed values of λx and Vapp according to Eq. 21.

20

( ) ( ) ( ) ( )

( ) ( ) ( )

1

,2

1

2

æö

---

ç÷

èø

+

=A+-+ =

éù

ëû

=A+-+

éù

ëû

iRx

ooo

iRxi

ooo

Ux i e

ie

o

qo

o

qo

j

ww

wj

ww

wwdwwdww

wdwwdww

( ) ( ) ( ) ( )

1

0, 2

==A+-+

éù

ëû

i

ooo

Ux i e

o

o

wj

w

wwdwwdww

( ) ( )

( )

2

22

21

++

Y= = = =

x

Sapp

iRxi ix

VV

iRx

xi

Ueee

Ue

o

q

o

q

o

o

wj

wl

w

w

w

wj

ow

w

ww

( )

Ψexp

V

æö

=ç÷

ç÷

èø

x

app

ix

w

w

14

3.6 Step 6: Generalized participation factor

Bi

: identification of the excited modes

1

For each mode i, (i=1,…,N), substituting Eq.16 in the decoupled Eq.5 (expressing the latter in the

2

frequency domain and with mode i normalized so that equals unity) results in:

3

(23)

4

where Qi(ω) is the Fourier component of qi. A harmonic solution for Qi is obtained by solving the

5

following equation:

6

(24)

7

where Ai is the dynamic amplification factor, and Bi is the generalized participation factor defined

8

as:

9

(25)

10

Note that Bi(ω) is a frequency dependent complex number for the case of non-uniform excitation,

11

or takes the value of the conventional modal participation factor Γi for the uniform excitation case.

12

Since Bi does not depend on the properties of the “actual” support time histories but on the model

13

used to represent the spatial variability of ground motion, the ratio:

14

(26)

15

can be used to quantify the potentially amplified contribution of the excited anti-symmetric modes.

16

This ratio was first defined by Price & Eberhard [4] who used it in the framework of a coherent

17

analysis in order to modify the modal participation factors of the anti-symmetric modes. In their

18

approach, the frequency dependence of Bi was ignored and it was only evaluated each time at the

19

natural frequency of the examined mode. In Eq.26, Bi,max is the upper bound of Bi, which is different

20

from the absolute value of modal participation factor Γi used under synchronous input motion:

21

(27)

22

M=T

φMφ

iii

( ) ( ) ( )

( )

( ) ( )

( )

22

2-

++=-

T

φMR Ψω

!! ! it

iiiiii o

QQQ Ue

w

wwzwww w w

i

( )

AB -

=it

iiio

QUe

w

w

( )

( )

( )

B=T

φMRΨω

i

w

i

( ) ( )

( )

ii

i,max

1

BB

B

=

=

åT-1

ic

φMK K

M

k

k

ww

( ) ( )

,max

11

B

==

G=¹=

åå

T-1 T-1

ic ic

φMK K φMK K

MM

ii

kk

kk

15

In the framework of the present simplified method, the ratio |Bi(f)|/Bimax is calculated for each

1

mode i (either symmetric or anti-symmetric) in a desired range of excitation frequencies (e.g. 0.2-

2

20Hz). Fig.3 illustrates indicative shapes of this ratio for the case of a symmetric (Fig.3 top) and

3

an anti-symmetric mode (Fig.3 bottom) and for different earthquake scenarios: (a) considering only

4

the wave passage effect (Fig.3 left), (b) considering both the wave passage effect and the loss of

5

coherency (Fig.3 middle), and (c) considering a perfectly uniform motion (Vapp→∞) (Fig.3 right).

6

These results correspond to the case of the Lissos bridge, which is subsequently presented in

7

Section 4 (with the symmetric and anti-symmetric modes corresponding to the 3rd and 4th ones cf.

8

Fig.8).

9

10

Figure 3. Indicative shapes of the |Bi(f)|/Βi,max ratio in the case of symmetric (top) and anti-symmetric

11

(bottom) modes for three different earthquake scenarios. The results refer to Lissos bridge (Section 4).

12

From Fig.3 it can be seen that spatial variability of earthquake ground motion significantly amplifies

13

the contribution of the anti-symmetric modes (highlighted regions in Fig.3 bottom) in the total

14

structural response and almost throughout the whole excitation frequency range, when compared

15

to the uniform excitation (red line). On the other hand, except for some narrow frequency ranges

16

that may arise (highlighted areas in Fig.3 top), it generally de-amplifies the contribution of the

17

symmetric ones. Comparing the left and middle parts of Fig.3 it can also be observed that, when

18

the loss of coherency is additionally taken into account, the |Bi(f)|/Bimax curve, despite retaining

19

its shape, is shifted towards lower frequencies. It is important to note that, as anticipated, the

20

|Bi(f)|/Bimax ratio matches the boundary|Γi|/Bimax ratio as the excitation frequency tends to zero

21

16

(Fig.3 left & middle) or as the Vapp tends to infinity while λx=0 (Fig.3 right). With respect to the

1

latter case of synchronous excitation, it could be expected that the ratio |Bi(f)|/Bimax would be

2

equal to zero given that, as it is well-known, anti-symmetric modes have no contribution to the

3

response under synchronous ground motion. However, this would only be valid in the case of fully

4

symmetric structures, something rarely met in real bridges. Since the results presented herein refer

5

to the non-symmetrical Lissos bridge (Section 4), the terms “symmetric” and “anti-symmetric” are

6

used in a wider scense. This is also illustrated in the modal participation factors of these modes

7

when considering uniform excitation; these are equal to Γ3=-8.67≠0 for the anti-symmetric and

8

Γ4=34.45≠0 for the “symmetric” mode. As such, the respective |Bi(f)|/Bimax ratios are calculated

9

as 0.103 and 0.662 (cf. Table 2 of Section 4.4).

10

3.7 Step 7: Frequency-dependent scale factor SFi(ω): quantification of the excited modes

11

Based on the results of Step 6, the scale factor (i.e., degree of amplification/deamplification)

12

for the modal participation factor of mode i due to multi-support excitation is given by

13

the ratio:

14

(28)

15

Fig.4 presents an example corresponding to the anti-symmetric mode of Fig.3 (bottom) (Step 6):

16

(or as presented in figures) represents the ratio of the blue (wave passage effect)

17

or yellow (wave passage effect & coherency) curves over the, straight, frequency-independent red

18

lines of uniform excitation shown in Fig.3. It can be observed that, in the case of asynchronous

19

ground motion, the scale factor corresponding to the anti-symmetric mode exceeds unity, having

20

peaks at specific frequencies (see section 5.3), while, as anticipated, under uniform excitation the

21

scale factor is SFi(f)=1 (Fig.4 right). This is effectively the key proxy used herein for higher mode

22

excitation due to SVEGM.

23

( )

i

SF

w

( ) ( )

,max

i i,max

BB

ΓB

=ii

i

SF

w

w

( )

i

SF

w

( )

i

SF f

17

1

Figure 4. Indicative shapes of the frequency dependent scale factor SFi(f) in the case of anti-symmetric

2

mode for three different earthquake scenarios (cf. Fig.3 bottom). The results refer to Lissos bridge (Section

3

4).

4

3.8 Step 8: Mean scaling factor across the frequency range of interest

5

Due to the fact that (or as presented in figures) is frequency-dependent, the need

6

arises to define a mean value that can be easily used for design purposes. As a result, a specific

7

frequency bandwidth needs to be defined. A reasonable assumption is to adopt the frequency range

8

that exceeds a certain level of the assumed PSD, similar to the one used in the case when partially

9

correlated motions are generated in a time history analysis framework. Herein, the bandwidth is

10

proposed to be measured at the level where the power of the spectrum equals ,

11

based on the definition of the respective bandwidth on the Fourier spectra of an accelerogram [76].

12

Fig.5 provides an illustrative example of the bandwidth definition in the case of the Clough &

13

Penzien PSD for firm, medium and soft soil conditions. Any other PSD (including an evolutionary

14

one) could be used instead. Estimation of the in the desired range is performed for each mode

15

i, {i=1,…N} but to be practical, only modes exhibiting >1 (i.e., detrimental amplification due

16

to SVEGM) are picked to be exploited in Step 9.

17

i

SF

( )

i

SF

w

( )

i

SF f

i

SF

( )

max

12PSD

i

SF

i

SF

18

1

Figure 5. Definition of the bandwidth in which the mean mode amplification scale factor is estimated

2

for the case of the Clough & Penzien PSD and for (a) firm: 0.86-2.94Hz, (b) medium: 0.955-1.83Hz, and

3

(c) soft: 0.655-0.865Hz soil (Der Kiureghian & Neuenhofer [11] parameters used for the spectra).

4

3.9 Step 9: Static analyses with lateral force patterns

Fi

5

Having identified (Step 6) and quantified (Steps 7-8) the higher modes that have can be excited by

6

spatially variable ground motions, a set of elastic static analyses is performed, for all modes with

7

with loads calculated by the following expression:

8

(29)

9

In Step 11, the response quantities derived from these static analyses (which take into account the

10

additional stresses imposed by the spatial variability of earthquake ground motion) are combined

11

with the respective ones obtained through conventional analysis under the assumption of uniform

12

ground motion (Step 10). In Eq. 29, the subtracted “-1” term essentially restricts the contribution

13

of these forces to be accounted only once in the total uniform response of Step 10.

14

3.10 Step 10: Conventional analysis for uniform ground motion

15

Any of the available methods for the dynamic analysis of bridges (e.g. response spectrum or linear

16

time history analysis) can be used to derive the uniform-excitation design quantities according to

17

the applied seismic code provisions. Special attention should be paid, though, for the case of time

18

history analysis; the peak response extracted should be used in Step 11.

19

3.11 Step 11: Bridge design quantities considering SVEGM effects

20

In this step, the response quantities derived from the conventional analysis of Step 10 (Mconv.) are

21

combined with the respective quantities (MFi) under asynchronous excitation estimated in Step 9

22

by applying the SRSS rule:

23

i

SF

1>

i

SF

( )

( )

2

1=-×G×× i

FMφ

ii iii

SF D

w

19

(30)

1

Overall, the method introduced herein focuses on the dynamic properties of the bridge as a proxy

2

of its potential amplification due to spatial variability. Currently, the assumption is made that the

3

soil is uniform, however, the mean soil properties can be employed when defining the PSD for the

4

case that ground conditions significantly vary along the bridge length. Since the method is based

5

on the natural modes of a bridge, it is essentially an elastic method. In the following, the accuracy

6

of the proposed method is verified for the case of two real bridges, against the predictions of

7

rigorous time history analysis using partially correlated seismic ground motions.

8

4. APPLICATION

9

4.1 Overview

10

Two real bridges in Greece were used as a test-bed to study the applicability and efficiency of the

11

proposed method for considering the impact of asynchronous excitation on the seismic response

12

of bridges. Uniform soil conditions of soil type A (EC8), maximum ground acceleration ag of 0.16g

13

and collectively, three seismic input motion scenarios were assumed: (a) synchronous excitation,

14

(b) asynchronous excitation due to finite wave propagation velocity Vapp=1000m/s, and (c)

15

asynchronous excitation due to partially correlated input motion with Vapp=1000m/s and λx=0.5

16

(dimensionless drop coherency parameter). It must be noted that values of Vapp and λx in cases (b)

17

and (c) are indicative and can be substituted with any other value within their valid range. The

18

effects of multi-support excitation were summarized through the impact mean ratios ‘ρ’, defined

19

as the maximum seismic demand at each pier (e.g. pier base bending moments) under

20

asynchronous ground motion, over the respective EDP under uniform input motion.

21

4.2 Bridge description

22

4.2.1 Lissos bridge

23

The Lissos River motorway bridge is an 11-span, base-isolated, R/C structure with an overall

24

length of 433m, located along the Egnatia Highway in northeastern Greece [77]. It consists of two

25

independent branches. The deck is a continuous, single-cell, pre-stressed concrete box with a

26

constant depth of 2.75m and 14.25m wide (concrete class B35 (kg/cm2), reinforcing steel class

27

St420/500, pre-stressing steel class St1570/1770), resting through elastomeric bearings on 10 piers

28

and roller bearings (450x600x55.5, with movement capacity +265mm/-365mm) on the abutments.

29

The expansion joints are 330mm.

30

22

.

M=M+M

åi

conv F

i

20

The transverse displacement of the deck over each pier is restricted to 10cm by stoppers of 1.20m

1

height, while it is prevented at the abutments through lateral elastomeric bearings. The piers are

2

made of reinforced concrete (class B35, reinforcing steel class St420/500) and their heights vary

3

between 4.50m to 10.50m. Fig. 6 illustrates the cross sections of the piers and the deck. This

4

particular bridge was adopted for study on the following grounds: (a) it is long enough for being

5

sensitive to to spatial variability, (b) it has been extensively studied for both synchronous and

6

asynchronous ground motion scenarios, and (c) it is insensitive to statically imposed displacements

7

which are applied in the framework of the simplified Eurocode 8 methods.

8

9

Figure 6: The Lissos River road bridge and its cross sections (dimensions units: meters).

10

4.2.2 Metsovo bridge

11

The ravine bridge of Metsovo (Fig. 7) is a 4-span R/C structure, consisting of two independent

12

branches with an overall length of 537m, located along the Egnatia Highway in northwestern

13

Greece. Holding the record of the tallest bridge of the Egnatia motorway (the tallest pier being

14

110m high), it was constructed with the balanced cantilever method. The deck is a continuous,

15

single-cell, limited pre-stressed concrete box, with a varying depth from 13.0m (at pier M2) to 4.0m

16

(in the key section) (concrete class B45) and a width of 13.95m. The superstructure is

17

monolithically connected to piers P2 (110m high) and P3 (35m high) (concrete class B45), resting

18

through pot bearings on pier P1 (45m high) (concrete class B45) and roller bearings on the

19

abutments. Piers P2 and P3 are founded on large circular Ø12m rock sockets in the steep slopes

20

of the Metsovitikos river ravine, at depths of 25m and 15m respectively. Fig.7 illustrates the cross

21

sections of the piers and the deck. In constrast to the Lissos bridge, this one was selected due to

22

its monolithical connection at the two pies.

23

21

1

Figure 7: The ravine Metsovo bridge and its cross sections (dimensions units: meters).

2

4.3 Numerical analysis

3

Finite element models of both bridges were developed in SAP2000 [78]. In these, the piers, the

4

deck and the stoppers were simulated with beam elements while the elastomeric bearings of the

5

Lissos bridge (equivalent linear properties at secant stiffness assumed) and the pot bearings of the

6

Metsovo bridge were modeled with link elements. Link elements were also used to simulate the

7

gaps between the deck and the stoppers; however, since the proposed approach is restricted to

8

linear response analysis, gaps were considered to be inactive (gap opening assumed to be infinite)

9

in order for geometrical non-linearities due to pounding to be excluded during the validation

10

process (THA). The base of the piers was considered fixed and the deck at the abutments was

11

assumed pinned along the transverse, and free to slide along the longitudinal direction. Fig.8 and

12

9 illustrate the eigenmodes with participating mass ratios greater than 1% along the transverse

13

direction Uy or around the vertical axis Rz for the Lissos and the Metsovo bridge, respectively.

14

Conventional analysis under uniform ground motion (as required in Step 10 of the proposed

15

methodology) was performed by means of the response spectrum analysis using the design

16

response spectrum (Type 1) of EC8 assuming a behavior factor q=1. This spectrum was also used

17

as the target one for the generation of partially correlated motion sets used in the validation process

18

(Section 5). In the latter case, multi-support excitation was applied to the bridge in the form of

19

displacement time histories; these were calculated from the respective synthetic accelerograms after

20

22

being subjected to the necessary baseline correction with the use of the appropriate, for each case,

1

second order polynomial. The Newmark’s method for direct integration (γ=1/2, β=1/4) was

2

employed to solve the motions’ differential equations, while energy dissipation was modeled

3

through Rayleigh damping.

4

5

Figure 8: Finite element model of the Lissos bridge and its eigenmodes with participating mass ratios greater

6

than 1% along the transverse direction Uy or around the vertical axis Rz.

7

8

Figure 9: Finite element model of the Metsovo bridge and its eigenmodes with participating mass ratios

9

greater than 1% along the transverse direction Uy or around the vertical axis Rz.

10

4.4 Application of the proposed method for Lissos and Metsovo bridges

11

Application of the proposed method for the two bridges was made following the 11 steps outlined

12

in Section 3. After obtaining the [NxN] mass M and stiffness K matrices, modal analysis was

13

employed to determine the structures' eigenperiods and eigenmodes, with those with participating

14

mass ratios greater than 1% along the transverse direction Uy or around the vertical axis Rz

15

illustrated in Fig.8 and Fig.9 for the Lissos and Metsovo bridges respectively. Considering only

16

transverse excitation, the [NxM] influence matrices R were computed. Since the kth column {rk}

17

23

of the influence matrix R corresponds to the structure's response under unit static displacement at

1

the kth support DOF (with all other support DOFs constratined), twelve static analyses for the

2

Lissos and five static analyses for the Metsovo bridge were performed. Modal participation factors

3

for each mode i associated with the kth support excitation were calculated by Eq. 15. The values of

4

Γi,k illustrated in Fig. 8 and 9 are respectively summarized in Tables 2 and 3. In these tables, the

5

last column is dedicated to the conventional modal participation factors Γi. These are computed

6

directly by SAP2000 [78] and are used to verify the accuracy of the Γi,k computation according to

7

.

8

Table 2: Modal participation factors Γi,k of the eigenmodes of Lissos bridge with participating mass ratios

9

greater than 1% along the transverse direction Uy or around the vertical axis Rz.

10

Mode

Γi,1

Γi,2

Γi,3

Γi,4

Γi,5

Γi,6

Γi,7

Γi,8

Γi,9

Γi,10

Γi,11

Γi,12

Γi

2

1.3

2.1

5.1

8.5

12.3

14.8

16.1

15.8

12.2

7.7

2.8

4.8

103.4

3

-10.1

-3.3

-7.1

-9.7

-10.1

-5.8

0.5

6.5

8.7

7.0

2.9

11.8

-8.67

4

13.3

1.8

3.3

2.9

0.4

-3.2

-4.1

-1.6

2.0

3.3

1.7

14.5

34.45

5

11.7

0.9

1.2

0.2

-1.3

-1.3

0.5

1.6

0.4

-1.1

-0.9

-13.2

-1.20

6

9.9

0.5

0.4

-0.4

-0.7

0.3

0.7

-0.3

-0.6

0.2

0.5

11.3

21.75

9

-7.9

-0.2

-0.1

0.3

0.1

-0.3

0.2

0.2

-0.3

0.0

0.2

8.9

0.97

Table 3: Modal participation factors Γi,k of the eigenmodes of Metsovo bridge with participating mass ratios

11

greater than 1% along the transverse direction Uy or around the vertical axis Rz.

12

Mode

Γi,1

Γi,2

Γi,3

Γi,4

Γi,5

Γi

1

-15.83

-0.41

-97.66

-47.65

13.64

-147.90

3

26.21

0.29

16.21

-47.99

10.63

5.36

6

-3.54

-0.06

2.55

-87.83

-20.99

-109.87

12

-0.60

-26.10

6.75

-2.67

1.97

-20.66

15

-8.57

-1.77

-39.64

1.61

-3.51

-51.89

20

7.52

-0.60

-10.38

-1.35

-7.69

-12.50

29

5.94

-0.32

8.11

1.87

5.07

20.68

30

-2.50

0.67

-24.07

-0.13

4.21

-21.82

46

2.95

-0.39

1.54

-14.51

-5.25

-15.67

47

-3.16

0.38

-1.24

-24.47

-0.75

-29.25

As already explained, three ground motion scenarios were defined: (a) synchronous excitation, (b)

13

asynchronous excitation due to finite wave propagation velocity Vapp=1000m/s, and (c)

14

asynchronous excitation due to fully-asynchronous input motion with Vapp=1000m/s and λx=0.5.

15

The non-dimensional spatial variability parameter Ψ(ω) and the generalized participation factor

16

Bi(ω) for each mode i were calculated using Eq.21 and Eq.25 in all three cases, for a range of

17

excitation frequencies between 0-20Hz.

18

For the effect of SVEGM on bridge response to be quantified, the ratios |Bi(f)|/Bimax were

19

computed, with Fig. 10 and Fig. 11 depicting them for the case of Lissos bridge under

20

asynchronous ground motion scenarios (b) and (c) respectively. The frequency-independent ratios

21

|Γi|/Bimax, corresponding to uniform ground motion scenario (a), are also shown in the same

22

,

1=

G=G

åM

iik

k

24

figures with a dashed line type. From the above figures, it can be concluded that the SVEGM

1

significantly amplifies the contribution of the anti-symmetric modes (right part of the figures) to

2

the total structural response almost throughout the whole excitation frequency range, while, except

3

for some narrow frequency ranges, generally de-amplifies the contribution of the symmetric ones.

4

Comparing Fig.10 with Fig.11 it can observed that, when the loss of coherency is additionally taken

5

into account, the |Bi(f)|/Bimax curve, despite retaining its shape, makes a shift towards lower

6

frequencies. In addition, as already discussed and is anticipated, the |Bi(f)|/Bimax ratio matches the

7

boundary|Γi|/Bimax one as the excitation frequency tends to zero or as the Vapp tends to infinity

8

while λx=0 (Fig.12).

9

10

Figure 10: |Bi(f)|/Bimax ratios of the modes for the Lissos bridge considering the wave passage effect

11

(Vapp=1000m/s and λx=0).

12

13

Figure 11: |Bi(f)|/Bimax ratios of the modes for the Lissos bridge considering both the wave passage effect

14

and the loss of coherency (Vapp=1000m/s and λx=0.50).

15

25

1

Figure 12: |Bi(f)|/Bimax ratio of the symmetric and anti-symmetric modes for the Lissos bridge considering

2

Vapp=500000m/s (time delay from Ab1 to Ab2 = 0.0009sec) and λx=0.

3

Scale factors SFi(ω) (or SFi(f) as presented in figures) for each modal participation factor of mode

4

i were then computed (Eq.28). Despite Vapp being equal to 1000m/s in the analyses, SFi(ω) were

5

additionally calculated in the Vapp range between 100-2000m/s. This would not be needed in an

6

actual design case, but is presented herein to facilitate better understanding of the key problem

7

parameters.

8

The higher mode scale factor SFi(ω) for the Lissos bridge under ground motion scenarios (b) and

9

(c) are respectively presented in Fig. 13 and Fig. 14 with cross sections of the surface graphs

10

illustrated in Fig. 15 and Fig. 16. In Fig. 13 and Fig. 14, the Vapp=1000m/s case is indicated with a

11

black line, while in Fig. 15 and Fig. 16, each colored line corresponds to a specific mode with the

12