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Important feature of calculating bridges under seismic action

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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.
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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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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
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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8
E3S Web of Conferences 157, 06020 (2020) https://doi.org/10.1051/e3sconf/202015706020
KTTI-2019
... Under seismic vibrations, the length of structures is one of the important factors affecting their seismic resistance. Research in this sphere given in [15], led to the development of a methodology for calculating multi-support structures, which in turn led to the need to change the existing methodology for calculating seismic resistance, considering the asynchrony of excitations of the structure supports. The calculation results obtained, show that an account for the asynchronous excitation of the bearing points of an extended system significantly reduces the inertial seismic loads on its elements. ...
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