TAKING ACCOUNT OF DAMPING IN ESTIMATING STRUCTURE

EARTHQUAKE STABILITY

К.V. Vorobyova1, О.P. Nesterova1, N.V. Nikonova1, А.М. Uzdin1*, М.Yu. Fedorova2

1St. Petersburg Transport University, Moskovsky Prospect 9, St. Petersburg, 190031 Russia

2St. Petersburg State University, Universitetskaya Nabereznaya 7-9, St. Petersburg, 199034, Russia

*e-mail: uzdin@mail.ru

Abstract. Methods of building damping matrixes for estimating earthquake structure stability

are considered. Modal expansion and eigenvalue problem taking into account the structure of

eigen vector matrixes are analyzed.

1. Introduction

In estimating structure earthquake stability engineers are increasingly faced with the problem

of taking damping into account. In most structures their members have different damping

parameters. For example, bridges are characterized by damping in rolling stock dampers, in

steel spans, in bearings, in concrete piers and in the base soils. Even for conventional

buildings at least energy dissipation in the building and the foundation soil should be taken

into account. However, current regulations do not considered damping at all.

In practice, the following tasks most frequently arise:

that of taking into account internal friction in structure material and in the

ground base,

that of taking into account the geometric dispersion of energy in the soil due to

energy radiation from the structure by elastic waves

that of energy dissipation in special damping devices.

2. Building the damping matrix and motion equation.

The questions of accounting internal friction in the material are well-studied [1-3 and others].

The most convenient way to take internal friction into account is to use the generalized

hypothesis by E.S. Sorokin [2] about the proportionality of damping matrices and that of

stiffness for the structure members. In this case, substitution of the modulus of elasticity Es of

the s-th structure element on the corresponding product sEs, makes it possible to obtain the

damping matrix BC in accordance with E.S.Sorokin hypothesis by instead of the stiffness

matrix R. Thus obtained the motion equation represented in complex form is the following:

**

c

*

0i YMqRBqM . (1)

Here complex values of generalized displacements and disturbances are marked by symbol *

[3], M is the inertia matrix of the system, R is the stiffness matrix.

Because of well-known disadvantages of equation (1) the internal friction model of

E.S. Sorokin is replaced by an equivalent viscous one [3].

KXХBB1

cэкв

, (2)

Materials Physics and Mechanics 26 (2016) 57-60 Received: November 1, 2015

© 2016, Institute of Problems of Mechanical Engineering

where X is the matrix of eigenvectors of the same system without damping (undamped

system), K2 = 1, 2,… n = is the diagonal matrix of the eigenvalues of the matrix M-1R.

Taking into account formula (2), we get a real system of equations equivalent to (1):

0экв YMRqqBqM . (3)

For systems with proportional damping, i.e., when the matrix M-1Вс has the same

system of eigenvectors Х as the matrix M-1R, equations (1) and (3) are expended into the

modes Tn21 ,..., XXΞq :

*

j

*2

jj

*

j pki1 j

2

jjjjj pkk . (4)

Here jjk ,2

j

j

jk

, pj are elements of the vector 0

1YX , j are the diagonal elements of

the matrix ХBMХ c

11 == 1, 2,…n.

If damping is not proportional, one can retain only the diagonal elements in the matrix

ХBMХ c

11 , considering ХBMХ c

11 1, 2,…n. Then we can get another view of the

viscous damping matrix which is proportional in according to its building:

ΓKXMХχXMХB11

пр

, (5)

where =1, 2,…n is the modal matrix of inelastic resistance coefficients.

Thus, the generalized hypothesis by Sorokin can be taken as the basis for building the

damping matrix, taking into account the internal friction in the material. It allows us to build

the matrix Вс and pass on to the abovementioned viscous damping matrices using formulas

(2) or (5). Calculations show that when s<0.3, which is typical of conventional materials and

soils, the use of formulas (2) and (5) give similar results [3].

Somewhat more complicated is the case with energy dissipation due to its geometric

radiation into the soil base. As shown by V.A. Ilichevin [5], this dissipation for vertical and

shear vibrations is frequency dependent and can be described for a wide range of frequencies

by classical viscous damping, i.e. it can be characterized by the coefficient of viscous

damping. As for rotary vibrations, here the dissipation frequency dependence is more

complicated [5], but in the acceptable range of frequencies it can be considered as frequency

independent and describes the ratio of inelastic resistance [6]. Within the framework of the

FEM geometric energy dissipation is taken into account along the border of the design base

area. In most cases, damping matrix is not proportional and its elements lead to the values of

far exceeding to the value of 0.3. As a result, the formulas (2), (3) and (5) may lead to

different results.

Limited possibility of application of formulas (2), (3) and (5) is typical when using

special dampers [7]. Some of them are directly described by viscous or hysteretic damping,

however, for a large part of dampers the resistance force is considered to be non-linear and is

described by dependence [7, 8]:

qsignqQQ 0

, (6)

where Q0 and are the damper parameters.

In the paper [8] it was shown, that nonlinear damping (6) can be represented with high

accuracy as the sum of the viscous and dry friction (DFD) dampers:

qsignFqbqsignqQQ экв0

. (7)

58 К.V. Vorobyova, О.P. Nesterova, N.V. Nikonova, А.М. Uzdin, М.Yu. Fedorova

Taking into account equation (7), the equation of oscillations can be written in the

following form

тр0экв QYMRqqBqM , (8)

where Qтр is the vector of generalized forces caused by friction in the open DFD and residual

displacement in the closed DFD.

3. The modal expansion of motion equations

An important problem for all forms of equations which take damping into account (1), (3),

and (8), is their modal expansion. If daqmping is due to internal friction in the material, the

approximate modal expansion using modes of the undamped system is permissible because of

the relatively small damping forces. In other cases, such modal expansion is not justified. In

this regard, in engineering practice methodologies and software tools of calculating complex

eigenvalues and vectors of the systems under consideration began to be used. Thus, it is

necessary to work with full matrices of systems (1) and (3), which have the form

For systems (3,8)

0E

RMBMA

11

For the system (1)

0E

BRM0A

1

ci

The widespread software being used in Russia, instead of MicroFe [9], does not

provided the solution of complex problem.

The authors of this paper have proposed two methods of determining complex

eigenvalues and vectors. The both methods are based on the initial approximation of

eigenvalues and eigenvectors х by identical ones of the undamped system.

At this initial approximation the complex eigenvector matrix can be written as

XX

ΩΩΓ

XΩΩΓ

XZ

**)0(22

ii, (9)

and the complex matrix of eigenvalues takes the following form

=2

,2

,2

,2

,2

,2

nn2211nn2211

+i

)n(

*

)2(

*

)1(

*

)n(

*

)2(

*

)1(

* ,,,,, , (10)

where 2

jj

)j(

* 25.01 .

The first method is based on the fact that approximations (9.10) are accepted to have a

slight error, which is iteratively corrected. At the i-th stage the eigenvalue and the

corresponding eigenvector z(i) are written as follows:

(i 1) (i)z z ( i ) ; ( i 1 ) ( i )

* * ; (11)

At the same time the following condition is to be met

(i 1) (i 1) (i 1)

*· · A z z . (12)

After putting (10) into (11) we obtain

( - ( i ) ( i ) ( i ) ( i ) ( i )

* A E Δ z Mp Δ , (13)

where (i) (i)· - · ( i )

*Mp A z z .

59Taking account of damping in estimating structure earthquake stability

If we neglect a small quantity · in equation (12),we obtain a system of n equations in

n + 1 unknowns (1), (2),… (n) and . The description of the algorithm of solving this system

is given in paper [10].

The second approach uses the well-known method by Leverie. According to it for the i-

th eigenvector the following condition takes place

(0)lim( )n

i in

z A z . (14)

Here (0)

iz is an initial approximation of the eigenvector in accordance with (9).

For systems with a sparse spectral range every initial approximation by formula (8)

ensures the convergence to the corresponding eigenvector. The situation gets more

complicated for systems having a tight specter range with points of condensation.

One can use mode correlation coefficients ij [12] as an empirical criterion of

applicability of the approximate calculating method

jkjk

2

j

2

j

2

k

2

k

2

j

2

k

2

jk

2

jk

jk

3

j

3

kjk

kj

kkkkkkkkkk

kkkk2

. (15)

If ij <0.2, one can ignore the influence of damping on oscillation modes.

References

[1] E.S. Sorokin, On the theory of internal friction in the oscillations of elastic systems

(Gosstroyizdat, Moscow, 1960). (In Russian).

[2] A.I. Tseitlin // Building Mechanics and Calculation of Structures 4 (1981) 33.

[3] A.A. Dolgaya, A.V. Indeykin, A.M. Uzdin, The theory of dissipative systems (Petersburg

Transport University, St.Petersburg, 1999).

[5] V.A. Ilichev, In: The dynamic analysis of structures for special impacts, Handbook

(Stroyizdat, Moscow, 1981), p. 114. (In Russian).

[6] Haibin Wang, I.O. Kuznetsova, A.M. Uzdin, U.Z. Shermukhamedov // Bulletin of Civil

Engineers 3 (2010) 91. (In Russian).

[7] A.M. Uzdin, S.V. Elizarov, T.A. Belash, Earthquake resistant costructions of transport

buildings and structures (Publishing House "Training Center for Education in Rail

Transport", Moscow, 2012). (In Russian).

[8] N.V. Durseneva, A.V. Indeykin, I.O. Kuznetsova, A.M. Uzdin, M.Yu. Fedorova // Journal

of Civil Engineering and Archtecture 9(4) (2015) 401.

[10] A.Yu. Soldatov, V.L. Lebedev, V.A. Semenov, In: VII Savinovsky readings. Abstracts.

(St. Petersburg Transport University, St. Petersburg, 2014), p. 21. (In Russian).

[11] D. Shchelkunov, A.M. Uzdin, A.A. Fedorov, M.Yu. Fedorova, In: VI Polyahovskie

readings (St. Petersburg State University, St. Petersburg, 2012), p. 86.

[12] A.A. Petrov, S.V. Bazilevsky, In: Earthquake-proof construction (domestic and foreign

experience) (TsINIS, Moscow, 1978), Series XIV, Issue 5, p. 23. (In Russian).

60 К.V. Vorobyova, О.P. Nesterova, N.V. Nikonova, А.М. Uzdin, М.Yu. Fedorova