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Important feature of calculating bridges under

seismic action

Lyubov Smirnova1

*

, Alexander Uzdin2, Natalia Polorotova2, and Maxim Freze3

1 JSC Research Center of Construction, 2-nd Institutskaya st., 6, Moscow, 109428, Russia

2 Emperor Alexander I St. Petersburg State Transport University, Moskovsky pr., 9, Saint-

Petersburg,190031, Russia

3 Transmost JSC, Podyezdnoy per., 1, Saint-Petersburg, 190013, Russia

Abstract. The purpose of the research is to show the main features of

calculating bridge taking into account the inhomogeneous acceleration field

along the structure length. The bridge is considered to be a linear structure

with point bearings on the soil base. For long bridges it is typical, that their

bearings are located in different seismogeological conditions. This result in

inhomogeneity of the acceleration field under the piers and non-synchronous

pier excitations. The motion equations of the system under consideration are

constructed and their decomposition into vibration modes is performed

without the account of external and internal damping in the system and with

the account of it. Based on the proposed decomposition, formulas for

determining seismic loads taking into account various seismicity under piers

are obtained. The result obtained show that the peculiarities considered can

be easily taken into account in existing software packages. As an example,

the authors analyzed the results of calculating a four-span beam railway

bridge. In calculating it was taken into account that the first and second piers

are located on sandstone, and the rest of them are on water-saturated loose

sand. The analysis showed that the account of the non-synchronous support

point excitation of the extended system reduces inertial seismic loads on its

elements significantly.

1 Introduction

Bridges have a number of fundamental features that significantly distinguish their work

during earthquakes from civil and industrial construction. Professor G.N. Kartsivadze made

a great contribution to studies of these features [1,2]. Thanks to his work in the former USSR

from the mid-70s of the last century the issues of seismic bridge resistance were singled out

in a separate section of the general theory of earthquake engineering. Over the past 40 years,

serious research has been devoted to the bridge seismic resistance. The issues of evaluating

seismic loads on bridges are considered in the monograph by G.S. Shestoperov [3]. A modern

analysis of the bridge seismic stability is given in the monograph [4]. One of the important

specific features of bridges is their length. In many cases such concept as the construction

site seismicity loses its meaning for the bridge.

*

Corresponding author: lyubovsmirnova80@gmail.com

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons

Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 157, 06020 (2020) https://doi.org/10.1051/e3sconf/202015706020

KTTI-2019

Usually, river bed piers are located in poor soil conditions, which may require an increase

of the design seismicity of the construction site per unit as compared with the area seismicity.

At the same time, bank piers can be placed on a bedrock, which reduces the design seismicity

by a unit. For longitudinal calculation of simple girder bridges G.N. Karzivadze substantiated

the ability to consider each pier independently [2]. He carried out similar studies for

transverse vibrations of bridges with a regular structure, in which stiffnesses of adjacent piers

differ slightly.

At present, bridges, especially road bridges, have become noticeably more diverse:

continuous, frame, cable-stayed bridges, viaducts over deep canyons and others are widely

used. The use of seismic isolating bearings becomes the basic solution of seismic protection

[4,5]. The calculation of such bridges should be done using a unified structural system taking

into account an inhomogeneous field of seismic excitation along the bridge length. In this

case, the bridge is considered as an extended linear structure with a point bearing on the base.

Lately, a lot of theoretical research has been devoted to calculating extended structures with

point bearing on the base. In Russia, these issues are considered in the monograph by Nazarov

Yu.P. [6], publications by Poznyak E.V. [7], Savvas P. et al. [8], Uzdin A.M. & Dmitrovskaya

L.N. [9], as well as in the abovementioned monograph [4]. Abroad, fundamental research of

the systems under consideration was carried out by A. Der-Kiureghian [10,11]. However, the

practical application of the theory of calculating extended multi-support structures has caused

certain difficulties so far. A description of these difficulties and ways of overcoming them

are discussed below on the example of calculating a specific bridge.

The paper considers the transverse calculation of a four-span beam railway bridge i.e.

displacements from the drawing plane are considered. The bridge is located in a seismically

dangerous area with seismicity 8 on the MSK scale. Piers 0 and 1 are located on sandstone,

which belongs to the first category soils and allows one to reduce the design seismicity by

unit. The remaining piers are located on water-saturated loose sand belonging to the third

category soils, which requires an increase in background seismicity by the unit. Thus, the

seismicity of the construction site for the first two piers is 7 degree on the MSK scale and for

the third, fourth and fifth piers are 9 degree on the MSK scale.

A schematic drawing of the bridge is shown in Figure 1. The design scheme of the bridge

is shown in Figure 2. To illustrate the features of calculating the bridge, taking into account

the inhomogeneous acceleration field along its length, a system with concentrated masses

with 12 degrees of freedom is considered. Moreover, the system has 5 bearing nodes

characterizing seismic input.

Fig. 1. Schematic drawing of the bridge

Fig.2. Design scheme of the bridge with concentrated masses

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E3S Web of Conferences 157, 06020 (2020) https://doi.org/10.1051/e3sconf/202015706020

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2 Materials and Methods

Let us write the initial oscillation equation for a heterogeneous excitation field along the

structure length. According to [21] it has the form

0E0I0

1

EI01

1

EI YβYβYCRBBYmCMRRYYBBYM )()(

(1)

Here M is the inertia matrix; Y is the displacement of the system nodes relative to the

position caused by the displacements of the bearing nodes of the weightless system; BI, BE

are matrixes of external and internal damping of the system; index I refers to internal nodes,

and E refers to external nodes; R is the system stiffness matrix; C is the reaction matrix in

the direction of the generalized coordinates from the displacement of the bearing nodes; m1

is the inertia matrix corresponding to the nodes at the boundary of the computational domain

of the base. This equation is somewhat different from those traditionally used in books [4,12]

and Russian Seismic Building Design Code SP 14.13330.2018.

First, in the equation right-hand side, the matrix of inertia before the vector Y0 is

multiplied by the matrix C and supplemented by a member m1. The presence of this member

is well known in structure dynamics [5]. In Russian literature on earthquake engineering the

presence of this member was indicated V.A. Semenov and A.G. Tyapin [13,14]. In finite

element calculations, when the inertia matrix M is determined by basic functions, such

member inevitably appears. To simplify the problem, we consider a computational model

with concentrated masses, i.e. we use the diagonal mass matrix.

Secondly, external and internal damping are separated in the equation. In this case, the

presence of external damping leads to the appearance of an additional member on the right

side of the oscillation equation. In this design model, there is no external damping. As shown

in [14],

0

I

1

IβCRB

, which also simplifies the equation.

Thirdly, pier excitations are accepted to be heterogeneous. It is determined by the vector

0

Y

. If all the supporting points are excited to one law and therefore, have one response

spectrum, but excitation amplitudes under each pier are characterized by their value

determined by the vector

I

ˆ

, then

0

ˆy

IY0

.

As a result, the resolving equation system gets the form

0

ˆy

ICMRRYYBYM 1

I

(2)

This equation coincides with the classical one if the vector Vp of the action projections

on the generalized coordinates directions is taken equal to

ICR 1ˆ

Note, that the proposed form (2) allows one to the calculate of a bridge when only one

support is excited. To do this, in the vector

I

ˆ

one must set the value corresponding to the

considered pier equal to 1, and set the remaining elements of the vector to zero. This

technique allows one to set its own impact and its response spectrum (dynamic curve) under

each pier.

Further we expand equation (2) according to the oscillation modes of a non-damped

system, i.e., we present solution (2) in the form

XΞY

(3)

Here X is the matrix of eigenvectors of the matrix M-1R, i.e. eigenvector matrix of a non-

damped system; = {1, 2, ... n} is the vector of principal coordinates.

The matrix X satisfies the well-known condition

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M-1RX = X = XK2, (4)

Where = K2 is the eigenvalues matrix of the matrix M-1R, K = k1, k2, ... kn is the

diagonal matrix of the eigenfrequencies.

Substituting (3) into (2) gives

0

1ˆy

ICMRRYΞXBΞMX I

.

After multiplying this equation by X-1M-1 and taking into account (4), we obtain

0

11211 ˆy

ICRXΞKΞXBMXΞI

(5)

In the general case, the matrix

XBMX I

11

is not diagonal, and the system is not divided

into independent equations. In the particular case when the matrices M-1B and M-1R have the

same system of eigenvectors, the equation system is divided [15]. In this case, damping is

called proportional. Adopting

χXBMX I

11

= 1, 2, … n, (6)

we get a divided equation system

0

112 ˆy

ICRXΞKΞχΞ

(7)

As a result, the system decomposes into independent equations of the form

0

2ydk jjjjjj

, (8)

where dj is a member of vector

ICRX ˆ

11

.

It is commonly accepted that j = jkj.

The quantity j has the meaning of the coefficient of inelastic resistance. We refer to these

coefficients as the modal damping, and the matrix Γ = 1, 2, ... n is called damping

spectrum.

To calculate complex systems, instead of inverting the matrix of eigenvectors, one can

use its orthogonality with weight

NMXX

T

(9)

where N = 1, 2, ... n is the diagonal matrix.

For the diagonal matrix of masses

n

iiijjmx

1

2

.

Seismic forces sij are calculated according to vibration modes using the well-known

formula

ijjjiij KTgAKms

)()(

1

(10)

where mi is the i-th system mass; A is the value of the base acceleration in fractions of

the gravity acceleration g; K1 is coefficient which takes into account permissible damage to

buildings and structures; (Tj) is the value of the response spectrum; K (j) is the coefficient

which takes into account the ability of buildings and structures to dissipate energy; ij is an

analogue of the participant mode factor.

For the mode participant factor ij, the well-known formula is used

k

n

k

k

pkkjij

ij

Vmxx

1

)(

(11)

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In contrast to the traditional formula for the mode participant factor, here

)(k

p

V

is an

element of the vector

ICR 1ˆ

.

ij

x

is the offsets in the i-th form in the direction of the generalized coordinate;

k

m

is the

inertial characteristics at the nodal point k.

If the acceleration field along the length of the bridge is one-valued and determined, then

we can restrict ourselves to calculating system loads caused by seismic forces and add them

to the loads from the support displacements. However, in fact, the situation is usually

different. The pier excitations are independent random functions. In this case, it is necessary

to carry out calculations for each pier excitations. This will result in many seismic force

matrices the number of which is equal to the number of piers. The matrix of seismic forces

from the excitation of the k-th pier will have the form

)(k

ijksS

. These forces will cause

strains, displacements and other factors the system behavior.

Let us denote the value of the r-th factor in mode j from the excitation of the k-th pier by

)(k

rj

, and the value of this factor from the displacement of the k-th pier

)(k

r

. Then the value

of the factor is determined by the following sum

2

1

1 1 1

)()(

2

)()(

n

k

Nf

j

Nf

pjp

k

rp

k

rj

k

r

calc

r

F

, (12)

where jp is the correlation coefficient of the j-th and p-th vibration modes. To calculate

the correlation coefficient, the formulas of Newmark [16] and A. Der-Kiureghian [10,11,12]

can be used. In Russia the formula of correlation coefficient was proposed by A.A. Petrov

(Petrov A.A., Bazilevskij S.V. Uchet vzaimnoj korrelyacii mezhdu obobshchennymi

koordinatami pri opredelenii sejsmicheskih nagruzok. – Ref.inf. «Sejsmostojkoe stroitel'stvo

(otechestvennyj i zarubezhnyj opyt)», seriya XIV, CINIS, M., 1978, vyp.5, p.23-28) and

widely used in engineering practice [17,18].

When only the correlation between the support excitation is taken into account,

expression (12) takes the form

2

1

1 1 1 1

)()()()()()()(

n

k

n

sks

Nf

j

Nf

j

s

rj

k

r

s

rj

k

rj

s

r

k

r

calc

r

F

, (13)

where ks is the correlation coefficient between excitations of the k-th and s-th pier. It is

believed that on non-rocky soils, correlation occurs when the distance between the bearing

points is less than 60 m [19]. In other cases, to estimate the value ks formulas based on the

hypothesis of a frozen wave are available [20].

3 Results

3.1 RSM calculation of a bridge with a homogeneous acceleration field

For a homogeneous acceleration field, one calculation of seismic loads was performed.

According to Russian Seismic Building Design Code SP 14.13330.2018 do not provide

setting different accelerations under piers, a synchronous pier excitations have been accepted

in the reserve, based on the design seismicity 9 on the MSK scale. To take account of

inhomogeneous damping, a method of distributing energy losses by vibration modes in

proportion to the mode energy was used. As is known, this approach result in a proportional

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damping matrix. The design seismic loads are calculated using the normative formula

(formula (12)). The calculation results are shown in table 1.

In table 1, the load values of dominating among the loads in the vibration mode (in the

matrix column) are mark out in a brighter type, and the cells of the table, in which are the

loads that determine the total load on the corresponding mass (in the matrix row), are mark

out in beige and yellow colors.

Table 1. Seismic mode loads for identical pier excitations

№

mass

Mode loads

Sum

mary

1

2

3

4

5

6

7

8

9

10

11

12

1

0.2

-2.0

0

2681

0

-57.9

-1.8

-6.8

52.8

-11.6

60.1

-1.4

2682

2

-0.0

0.3

0

-116.3

0

-394.6

-13.6

-60.6

1744

-266.4

1509

-43.2

2359

3

28.4

325.4

-0.8

51.9

0

1523

45.1

162.7

1011

9.6

212.4

-17.5

1877

4

0.1

-0.3

0

-1.6

-0.1

-863.0

-10.9

39.2

3768

374.3

-1065

-31.4

4027

5

-36.2

-263.2

-0.6

0.8

1.1

2153

1.9

-228.5

1385

-21.4

-310.7

0.9

2603

6

-0.1

-0.4

0.2

0

-3.5

-844.2

13.5

4.8

3454

-569.2

105.7

46.2

3603

7

0.3

7.6

-309.0

0

48.6

1599

152.1

152.1

946.8

37.8

247.5

17.4

1908

8

0

0

-0.3

0

-145.5

-320.0

-44.5

-44.;6

1095

910.7

902.6

28.3

1723

9

0

0

0.3

0

2632

-56.8

-5.9

-5.9

43.7

49.3

45.0

1.2

1634

10

-52.4

600.3

1.5

31.7

0

677.6

72.0

72.0

433.7

4.0

92.7

-7.4

1013

11

66.7

485.3

1.1

0.3

0.5

958.4

-101.1

-101.1

594.5

-9.0

-134.7

0.4

1241

12

-0.6

-14.0

570.3

0

22.4

710.3

67.3

67.3

405.6

15.7

106.6

7.5

1006

3.2 RSM calculation of the bridge taking into account the heterogeneity of the

acceleration field

To illustrate the significance of the effect of heterogeneity of the acceleration field along the

length of the bridge, we restrict ourselves to estimating the seismic load using formulas (10).

Note that the system under consideration is statically determinable and no stresses caused by

mutual pier displacements arise in it. Since the system has 5 bearing nodes, 5 matrices of

seismic forces from each support excitation were obtained. For these calculations the PGA is

assumed to be 1 m/s2 for the first two piers, which corresponds to the construction site

seismicity equal to 7 degree on the MSK scale, and the response spectrum (dynamic curve)

is adopted for the rock base. At the same time, for the next 4 piers, PGA = 4m/s2, which

corresponds to the construction site seismicity equal to the 9 degree on the MSK scale, and

the response spectrum (dynamic curve) is accepted for a weak base. As an example, tables 2,

3 and 4 show the matrices of seismic forces according to the oscillation modes from different

excitations under piers. Seismic forces caused by synchronous excitations with different

amplitudes under piers are shown in table 2. Seismic forces caused by nonsynchronous

excitations of the second and third piers are shown in tables 3 and 4. The conclusive

calculation results are given in table 5.

Table 2. Seismic mode loads for synchronous pier excitations with different amplitudes under piers

№

mass

Mode loads

Summary

1

2

3

4

5

6

7

8

9

10

11

12

1

-0.4

0.6

0

4.7

0

44.7

-27.7

-6.6

-38.7

2.2

-1.8

29.7

72.2

2

0.1

-0.1

0

0.2

0

304.4

-213.1

-59.6

-1278

50.8

-46.9

869.1

1592

3

-64.3

106.4

0.5

-0.1

0

-1175

708.1

160.0

-741.5

-1.8

-6.6

344.5

1610

4

-0.2

0.1

0

0

0

665.8

-171.5

38.5

2762

-71.4

33.06

630.4

2917

5

82.0

86.1

0.3

0

0

-1661

29.1

-224.7

-1015

4.07

9.6

-17.4

1964

6

0.1

0.1

-0.1

0

0

651.2

211.6

4.7

-2533

108.6

-3.3

-928.4

2785

7

-0.8

-2.5

170.7

0

0

-1233

-695.8

149.6

-694.1

-7.2

-7.7

-349.8

1631

8

0

0

0.2

0

0

246.8

163.1

-43.8

-802.5

-173.8

-28.0

-568.5

1043

9

0

0

-0.2

0

0.5

43.8

25.3

-5.8

-32.0

-9.4

-1.4

-25.02

65.83

10

118.5

-196.3

-0.8

-0.1

0

-522.7

314.2

70.8

-318.0

-0.8

-2.9

148.6

743.5

11

151.1

-158.7

-0.7

0

0

-739.3

12.4

-99.5

-435.9

1.72

4.2

-7.2

891.5

12

1.4

4.6

-315.0

0

0

-548.0

-308.5

66.2

-298.1

-3.0

-3.3

-150.7

781.4

Table 3. Mode seismic loads with excitation of pier No. 2

№ mass

Mode loads

Summary

1

2

3

4

5

6

7

8

9

10

11

12

1

0.1

0.2

0

-6.

0

4.1

10

6.6

-3.7

0.2

-5.6

-8.67

17.9

2

0

-0.1

0

0.3

0

27.9

76.9

59.7

-123.6

3.8

-139.6

-253.4

330.5

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3

16.2

25.7

0

-0.2

0

-107.4

-255.8

-160.5

-71.7

-0.2

-19.7

-100.4

345.4

4

0.1

0.1

0

0

0

60.9

61.9

-38.7

-267.2

-5.3

98.5

-183.8

351.9

5

-20.6

20.8

0

0

0

-151.9

-10.5

225.3

-98.2

0.3

28.7

5.1

292.1

6

-0.1

0.1

0

0

0

59.6

-76.5

-4.7

-244.9

8.1

-9.8

270.7

377.9

7

0.2

-0.6

0.4

0

0

-112.8

251.4

-150.0

-67.1

-0.5

-22.9

101.9

337.4

8

0

0

0

0

0

22.6

-58.9

43.9

-77.6

-12.9

-83.5

165.7

215.8

9

0

0

0

0

0

4.0

-9.2

5.8

-3.1

-0.7

-4.2

7.3

14.6

10

-29.8

-47.5

0

-0.1

0

-47.8

-113.5

-70.9

-30.8

-0.1

-8.6

-43.3

162.0

11

38.0

-38.4

0

0

0

-67.6

-4.5

998

-42.2

0.1

12.5

2.1

139.3

12

-0.4

1.1

-0.8

0

0

-50.1

111.5

-66.4

-28.8

-0.2

-9.9

43.9

148.9

Table 4. Mode seismic loads with excitation of pier No. 3

№

mass

Mode loads

Summary

1

2

3

4

5

6

7

8

9

10

11

12

1

-0.5

0.5

0

0

0

23.5

1.6

-37.8

-20.5

-1.4

33.1

2.0

59.202

2

0.1

-0.1

0

0

0

160.3

11.9

-340.1

-678.5

-32.1

830.9

58.6

1139

3

-80.4

81.0

0

0

0

-618.4

-39.6

913.8

-393.5

1.2

116.9

23.2

1184

4

-0.2

0.1

0

0

0

350.5

9.6

220.1

-1466

45.0

-586.3

42.5

1633

5

102.4

65.5

0

0

0

-874.5

-1.6

-1283

-538.9

-2.6

-171.1

-1.2

1657

6

0.2

0.1

0

0

0

342.9

-11.8

27.0

-1344

-68.5

58.2

-62.6

1392

7

-0.9

-1.9

3.1

0

0

-649.4

38.9

854.4

-368.4

4.5

136.3

-23.6

1144

8

0

0

0

0

0

129.9

-9.1

-250.2

-425.9

109.5

497.0

-38.32

722.2

9

0

0

0

0

-0.2

23.1

-1.4

-32.9

-16.9

5.9

24.8

-1.7

50.6

10

148.1

-149.5

0

0

0

-275.2

-17.6

404.2

-168.8

0.5

51.0

10.0

561.1

11

-188.7

-120.8

-0.1

0

0

-389.2

-0.7

-568.1

-231.3

-1.1

-74.2

-0.5

763.8

12

1.7

3.5

-5.7

0

0

-288.5

17.3

377.9

-158.2

1.9

58.7

-10.2

504.9

The calculation data as compared with the calculation data for the synchronous pier

excitations are shown in table 5.

Table 5. Seismic load calculation results

№ mass

Modal forces and design loads

For synchronous pier

excitations

For nonsynchronous pier excitations

Identical

Different

N1

N2

N3

N4

N5

Design loads

1

2682

73.2

1.793

17.918

59.202

68.37

2.848

92.259

2

2359

1592

19.347

330.538

1139

1381

70.312

1821

3

1877

1610

6.413

345.357

1184

1362

23.847

1837

4

4027

2917

15.01

351.99

1633

1397

65.391

2179

5

2603

1964

5.622

292.085

1657

1138

20.493

2031

6

3603

2785

11.956

377.904

1392

1501

77.134

2083

7

1908

1631

6.505

337.438

1144

1350

24.382

1801

8

1723

1043

13.162

215.751

722.16

910.249

112.92

1187

9

1634

65.8

0.668

14.622

50.581

60.819

12.452

81.404

0.668

10

1013

743.5

2.807

162.017

561.145

601.711

10.458

838.634

2.807

11

1241

891.5

2.46

139.271

763.842

502.833

8.986

925.083

2.46

12

1006

781.4

2.837

148.991

504.969

666.873

10.647

849.725

2.837

4 Conclusions

The non-synchronization of pier excitations when calculating seismic loads is easily taken

into account using existing software tools for the structure dynamic calculation. To do this,

it is necessary to modify the calculation of the mode participant factor included in the formula

for seismic loads and to calculate the set of seismic force matrices from the excitation of each

of the supports separately.

Certain difficulties can be connected with the summation of modal forces and forces

caused by pier excitations. If the vibration modes and pier excitations are statistically

independent, the calculated force can be estimated as the root of the sum of the squares of its

components. In more complicated cases, it is necessary to take into account the mode

correlation and the correlation of pier excitations. If the assessment of the vibration mode

correlation does not present any problems and is included in many software systems, the

correlation coefficients of pier excitations still require additional research.

In the considered example, when the system is statically determinable and the mutual

displacements of the supports do not cause additional forces in the system elements, taking

into account the inhomogeneous acceleration field along the structure length results in a

significant decrease of the design seismic loads.

7

E3S Web of Conferences 157, 06020 (2020) https://doi.org/10.1051/e3sconf/202015706020

KTTI-2019

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E3S Web of Conferences 157, 06020 (2020) https://doi.org/10.1051/e3sconf/202015706020

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