kholmyansky, 15 ecsmge .pdf

8/10/2019 Kholmyansky, 15 ECSMGE .pdf

1/9

Poceedngso he

5h

Euopean

ConerenceonSol Mchancsand

Goechnca Engneerng

GeotechnicsofHard Soils WeakRocks

Compes Rendus

du

15emCongres

Europeen

deMcanquedes Sos deGeoechnque

LaGeotechniquedesSolsIndures

Roches Tendres

Parti

Editedby

AndreasAnagnosopouos

National Technical

University

ofAthens

Mchael Pachakis

OTMConsulting EngineersSA

and

ChristosTsatsanifos

PANGAEA Consulting Engineers

LTD

S

r ss

Amsterdam

Berlin

Tokyo

Washington

DC


2/9

Proceedings of

th e

15th European

Conference on

Soil Mechanics

an d

Geotechnical Engineering

A. Anagnostopoulos et al. Eds.)

IO S

Press, 2011

2077

The authors and IOS Press. All rights reserved.

doi:10.3233/978-l-60750-801-4-419

Vbrational reliability

o

rigdstructures

on

sol

wthrandomelasticparamters

Lafiabilitedevbrationdestructures rigdessur le solavec

paramtres

elastiques aleatoire

M.L. Kholmyansky

1

NIIOSP

Research Institute

ofPJSC

Research Centre Civil Engineering

419

ABSTRACT

A probabilistic problem

of

vibrational reliability

of

structures

on the

soil with significant uncertainty

is

stated.

For the

simplest

dynamic models the dependence of the response on the level of the elastic stiffness is investigated. The problem s of reliability

determination for the deterministic and random loads are solved. The n um erical results are obtained and analyzed.

RESUME

Unprobleme

de

fiabilite probabiliste

de

vibration

de

structures

sur le sol

avec

un e

grande incertitude

e st

indique. Pour le s plus

simples des

m odeles dynamiques

de la

dependance

de la

reponse

au niveau de la rigidite elastique est etudie. Les problemes de

determination

de la fiabilite

pour

les

charges

deterministes et aleatoires

sont

resolus. Les resultats numeriques

sont obtenus

et

analyses.

Keywords: Deterministic dynamic models, random system parameters, probabilistic model, failure probability, reliability, ma-

chine foundations, soil elasticity

1 INTRODUCTION

Uncertainty plays a significant role in geotechni-

cal

engineering [1].That is particularly true for

dynamic

problems [2] because soil dynamic pa-

rameters

are

often leftundetermined

in

course

of

geological survey. Generally, the only way to ob-

tain

the

parameters

for the

vibration calculation

is

using

ofcorrelation

dependencies

for dynamic

parameters.Thismay lead to significant errors.

Uncertainty is usually accounted for using

partial

safety factors

for soil parameters [3, 4].

Nevertheless, in the problems of vibration calcu-

lation the selection of input parameters on the

Corresponding Author.

safe

side isgenerally impossible.For example,

reducing

the

soil

stiffness

in the

calculation

may

lead to both increase and decrease of the ampli-

tude; in the latter case vibration level underesti-

mation is possible that is non-conservative. Pro-

viding reliability through sound allowance

for

uncertaintyinstructural vibration calculationre-

quires

the

development

of new

methods

ofcalcu-

lation.

Probabilistic approach to uncertainty is used

as the most developed in geotechnical engineer-

ing [5, 6]. It

consists

in

representing soil parame-

ters by random variables having

specific

distribu-

tion laws instead of taking soil variability into


3/9

4

M.L.

Kholmyansky

/

Vibrational eliability

of

igid Structures

on

Soil

account

by introducing partial safety

factors

for

soil parameters. Some publications

basedon

that

approach

are

reviewed

in

[2]. This approach

de

scribes

reliable operationby the condition of re

liability [7], when the norms are set for probabil

ity of

failure

or of

reliable

operation.

The paper is devoted to studying of failure

probability dependence of soil random

stiffness.

In

necessary cases dynamic load amplitude is

supposed a random variable. The structure is

supposed rigid that

is

made rather

often, for ex

ample in vibration calculation of massive and

walllike machinefoundations[8].

To achieve the stated goal the

first

task is es

tablishing

a possibly simple

dependence

system

response soil

stiffness

level

in deterministic

case.

The second task is the probabilistic problem

statement. Due to theshortageofdatait is expe

dient to choose the simplest models. Then the

techniques for

probabilistic problem solutions

are developed, solutions obtained and analysed.

DEPENDENCE

OFRIGID STRUCTURE

VIBRATIONAL RESPONSE

ON

SOIL

STIFFNESS LEVEL

2.1

General equations

The matrix equation of small vibrations of the

soilstructure system (a rigid body on a viscoe

lastic

soil

mass) reads as

follows:

Mq Bq Kq = Q

(1)

whereq and Q arecolumn vectorsofgeneralized

displacements and forces, M, and are the

matrices

of

inertia, damping,

andstiffness of

soil

correspondingly.

Generally

the

system

has 6

degrees

of

free

dom

(3translational and 3rotational displace

ments). In

case

of

symmetry instead

of

vector

equation

(1) several

equations

for

vectors

of

lower dimension (or for scalars) may be derived;

the code

[8]

contains some solutions

for the

case

of harmonic excitation.

In

case

of

harmonic dependence

of

excitations

andresponses

on

time t

with angular frequency

,

i.e. Q = Pexp(z otf), q = U

exp icot)

one ob

tains

(2)

Under

linear dependence

of all the

compo

nents of generalized force vector on one scalar

value (Q

=/g,

/=

F exp(z fttf)

andwith singleob

served quantity linearly dependent on general

ized displacement vector u

=l

x

q ,

the expres

sion = A exp icot) is derived. The complex

amplitudeA is determined using the scalar trans

fer function:

A =

= H co)F co) ;

3

the expression for the transfer

function

(imped

ance) may be

found

in

[9]:

4

2.2

The

single parameter

of

soil

stiffness

Only simple dynamic models

from

the code [8]

are

considered below. They imply

the

propor

tionality

of all thecomponentsofsoil

stiffness

to

the

main elastic characteristic

of soil for spread

foundations coefficient ofelastic subgradere

action

C

z

C.

Elastic soil parameters determine not only

system

stiffness

but its

damping

also.In

dynamic

models [8] it is supposed

that

damping ratios are

dependent

only on foundation

inertial

parame

ters. This impliesthatdamping matrix is propor

tionalto

C

v

\

Therefore

A

=\

T

CK

(5)

where the matrices BOand

K

0

do not depend on

C=C

Z

.

Itiseasily

established

that

the

complex ampli

tude A is equal to the ratio of two homogeneous

polynomials of and ~

/2

; the

degree

of the nu


4/9

M.L.Kholmyansky /

Vibrational eliability

ofRigid

Structures

on

Soil 421

merator polynomial

is(22),and thedegreeof

the

denominator polynomial

is

2n;

n

amount

of

system

degrees of

freedom, i.e.

the

order

of

the matrices M, and K. In case of single de

gree

of

freedom

system the complex amplitude is

the

inverse

of the homogeneous polynomials of

thesecond

degree

ofand C*

/2

.

(9)

i.e.

sufficiently

large probability

of no

failure.

The

equivalent condition

of smallness of

failure

probabilityis

(10)

2.3 Complex dynamic stiffness of the

system

Instead of

using

impedance,

its

inverse value,

scalar dynamic

stiffness

D =

D C,co) may be

considered:

=

FID C,co);

(6)

itis acomplex quantity.

The most important variable describing sys

tem

response

is the real amplitude

a

\A. It is

found

by the

equation

a= F/\D C,co)\.

7

The

dynamic

stiffness is the

ratio

of a

homo

geneous polynomial

of

degree

2n

to a

homoge

neous polynomial

of

degree(22)

of C

/2

.

Hence

follows that

for

large

values

of

dynamic

stiff

ness along with itsmodulusare is asymptotically

proportional to C.

Henceforth we do not take into account the

phaselags,

suppose

thatthe force amplitude F is

realandconsider onlythereal amplitudea.

3

PROBABILISTIC PROBLEM

STATEMENT

AND

MAIN FORMULAE

3.1

Sufficient reliability condition

Thecommon conditionof no failure [8]reads as

follows:

a


5/9

422

ML.Kholmyansky / Vibrational Reliability

of

Rigid Structures on Soil

ter C,

determining

the stiffness

matrix. Earlier

in[10] uniform

and normal

laws

of distribution

were adopted.

Due to some limited experimental data

[11]

it

may be

supposed that

the

soil elastic

stiffness

(affecting

the vibration amplitudes) is

log

normally distributed random variable (i.e.with

normally

distributed logarithm) with the

coeffi

cient of

variation close

to 0.3

[9].This distribu

tion

law corresponds to elastic stiffness deter

mining

using

its

correlation

[8] to

soil

deformationmodulus.

3.5 Sufficient reliability condition in terms of

loads

The

condition

(8) may be

written down

in an

equivalent

form

[12, 2]

using

(6):

F


6/9

M.L.

Kholmyansky

/

Vibrational Reliability

of

Rigid Structures

on

Soil

4 3

F

i,=

0 a )

()

, a)

U

2

c)

c

Figure

2.

Special cases

fo r

fai lure intervals

of the stiffness

axis deterministic loads and random

stiffness

for single de

gree

of

freed om systems) :

a)L \= 0, for

zero

stiffness th e

dynamic

load exceeds

th e

l imiting value;

b)n

s

= 0, i.e. fo r

an y

s t i f f n e s s

t he

load

is

lower than

the

l imiting value;

c)general case.

where

pc C)

is the probability density function

for

lognormal distribution of C:

Pc C)

=

1

(7/2;

exp

2(T

2

(18)

The distribution parameters and // define the

stiffness expectation Co and coefficient of varia

tion

CV:

=

^

_ u a / 2

(19)

Since for large values of the dynamic

stiff

ness (and consequently the limiting load) is as

ymptotically proportional to C, the argument of

exp

function

in

(17)

is

proportional

toC

2

and the

integral converges rapidly.

5 NUMERICAL RESULTS AND THEIR

ANALYSIS

4.2

Random

load

In

case

of

dynamical action

of

machines with

ro

tating parts the load amplitude F in accordance

with the adopted Rayleigh law is supposed to

have the following cumulative distribution func

tion:

7H (16)

In

this equationF

0

> 0; F is themathematical

expectation of the random load amplitude

F;

co

efficient ofvariation of F equals0.523.Theran

dom load

is

supposed stochastically independent

of

soil stiffness.

Failure probability

in

this case

is

given

by the

following integral:

P =

f

(17)

5.1 Deterministic load

The simplest case is considered when the system

has a single degree of freedom. Figure 3 shows

the familyofdomains fordimensionless parame

ters

that

provide reliable operation or

failure

for

the

various levels of desired limiting probability

of

failure

P

u

under deterministic dynamic loads.

The

dimensionless parameters

are

F/ a

u

K

)

),

Q

= C D C O Q

and the damping ratio

; KQ

is the sys-

tem

stiffnessmathematical expectation

and G J

O

corresponding

system

non-damped

frequency.

ForPf>P

u

we

have failure domain,

and for

Pf P

u

the domain ofreliableoperation.The

coefficient

of variation ofsoil

stiffness

CV=

0.3.

The load growth expectedly

lowers

the

reli-

ability. For

high damping ratio

(g= 1) the

fre-

quency growth always causes the

reliability

growth meaning

that resonant

phenomena are

absent), and for small and

medium

damping

ra-

tios

thedependenceof

reliability

on

frequency

is

non-monotonic.


7/9


8/9

M.L. Kholmyansky / VibrationalReliability

of

Rigid Structures onSoil

42 5

5.2 Stiffness variability effect

Stiffness

variabilityeffect

is of

certain interest.

Figure 4 shows different reliability levels for

lowered

and

increased

coefficients

of

variation

of

soil(CV=0.1 and

CV=

0.5); damping ratio

f

is

taken 0.3.

The results of the calculations show that

change

of the

coefficient

of

variation

for

soil

stiffness

do not influence

qualitatively

the

gen-

eral

dependence of reliability on system parame-

ters.

Forsmallstiffness variabilitythe

interface

be-

tween domain

offailure and

domain

of

reliable

operation moves insignificantly with limiting

probabilityoffailurechange.

5.3 Random load

Figure

5

shows

the

results

for

random loads

in

case

of

system with single degree

offreedomfor

the coefficient of

variation

of

soil

stiffness

hav-

ing standard value of 0.3; in this

case

It is

seen

fromthe

results that except

for

large

damping

=

1) the

least adoptable load expec-

tancy corresponds to certain non-zero

frequency

(resonance). The analogous phenomenon was

observedfor thedeterministic load.

6 CONCLUSIONS

For

examining vibrational

reliability of

rigid

structures on

soil

it is

necessary

to

consider inde-

terminacy

of

elastic soil properties. That purpose

may be

obtained

by

using probabilistic problem

statement with simplest dynamic models

of

soil-

structure

system.

The paper contains the analysis of dependence

ofsystem vibrational behaviour

on the

single

pa-

rameter

of

stiffness

coefficient of

elastic

sub-

grade reaction (or some other, say soil elastic

modulus).

Time harmonic loads

of two

types

are

considered with deterministic amplitudes and

withrandom amplitudes distributed according

to

the

Rayleigh law.

Figure

5 .

Domains

of

failure

and

reliable operation

fo r

single

degree

of

freedom system with random R ayleigh load

fo r

dif-

ferent

levels of limiting probability of failure:

0.01;

0.25;

0.50; 0.75; 0.80; 0.85; 0.90; 0.95; 0.98

an d

0.99.


9/9

426

M.L. Kholmyansky / Vibrational

Reliability ofRigid S tructures

on

Soil

For the

both types

of

load closedformformu-

lae are derived, calculations performed and their

results analysed.

Theresults foundprovide thepossibility of

more sound decision making when designing

with required reliability;

fore

some cases eco-

nomic

benefit

may be obtained by removing un-

necessary reserve.

REFERENCES

[1] F.Nadim, Toolsa nd Strategies fo r Dealing with Uncer-

tainty in

G eotechnict,

Probabilistic Methods in Geo-

technical Engineering

eds.

D.V. G riffiths,

V.A.

Fen-

ton),

2007, 71-95.

[2 ] M.L. Kholm yansky, Dynam ic soil-structure interaction

considering random soil properties, Proceedings of the

12th International

Conference of

IACMAG,

2008,

2704-2711.

[3 ]

G O S T

27751-88, Reliability

of

constructions and foun-

dations. Principal rules of th ecalculations,

IzdateFstvo

Standartov,

Mo scow, 1989. Soviet Standard;

in

Rus-

sian).

[4 ]

EN

1997-1,

Eurocode7 - Geotechnical design, Part

1:

General rules,

European Committee

fo r

Standardiza-

tion,Brussels,2004.

[5] N.N. Yermo laev, V. V. Mikheev,

Reliability of struc-

ture

foundations,

Stroyizdat, Leningrad , 1976.

In

Rus-

sian).

[6 ] V.I. Sheinin, Y u. V. Lesovoi, V.V. Mikheev,

N.B. Popov, An

approach

to

reliability assessment

in

engineering calculations

of foundation

beds,

Soil M e-

chanics and Foundation Engineering,

27 1990), 32-

36 .

[7] V.V. Bolotin,

Random vibrations

of

elastic systems,

Ni-

jho f ,

Hag ue, 1984.

[8 ] SNIP 2.02.05-87,

Foundations

fo r

machines under

dy -

namic loadings,

TsITP, Moscow, 1988. Soviet Build-

ing Code; in R ussian).

[9 ]

M.L.

Kholmyansky, Vibration calculation of founda-

tions

fo r

machines with random dynamic loads

as

sys-

tems with random param eters ,Earthquake Engineering,

1998,

6-8. In

Russian).

[10]

A.I.

Tseytlin, N.I. G useva,

Statistical methods for cal-

culations of structures under group dynamic loading,

Stroyizdat, Moscow , 1979. In Russian ).

[11]

D.D.

Barkan, Yu.G . Trofimenkov, M.N. G olubtsova,

Relation between elastic and strength characteristics of

soils, Soil Mechanics and Foundation Engineering,

1974),51-54.

[12] M.L. Kholm yansky, Probabilistic meth od of calculation

of

machine

foundations under

periodic

loading,

Proc.

Russian

Conference

on Soil

Mech.

and Foundation

Engng, 4

1995),

669-674.

In Russian).

kholmyansky, 15 ecsmge .pdf

Documents