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Poceedngso he
5h
Euopean
ConerenceonSol Mchancsand
Goechnca Engneerng
GeotechnicsofHard Soils WeakRocks
Compes Rendus
du
15emCongres
Europeen
deMcanquedes Sos deGeoechnque
LaGeotechniquedesSolsIndures
Roches Tendres
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Editedby
AndreasAnagnosopouos
National Technical
University
ofAthens
Mchael Pachakis
OTMConsulting EngineersSA
and
ChristosTsatsanifos
PANGAEA Consulting Engineers
LTD
S
r ss
Amsterdam
Berlin
Tokyo
Washington
DC
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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
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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
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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
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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
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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.
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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.
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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:
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European Committee
fo r
Standardiza-
tion,Brussels,2004.
[5] N.N. Yermo laev, V. V. Mikheev,
Reliability of struc-
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foundations,
Stroyizdat, Leningrad , 1976.
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[6 ] V.I. Sheinin, Y u. V. Lesovoi, V.V. Mikheev,
N.B. Popov, An
approach
to
reliability assessment
in
engineering calculations
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beds,
Soil M e-
chanics and Foundation Engineering,
27 1990), 32-
36 .
[7] V.V. Bolotin,
Random vibrations
of
elastic systems,
Ni-
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Hag ue, 1984.
[8 ] SNIP 2.02.05-87,
Foundations
fo r
machines under
dy -
namic loadings,
TsITP, Moscow, 1988. Soviet Build-
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[9 ]
M.L.
Kholmyansky, Vibration calculation of founda-
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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-
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Stroyizdat, Moscow , 1979. In Russian ).
[11]
D.D.
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Relation between elastic and strength characteristics of
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periodic
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