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AVOIDANCE STRATEGY FOR SOIL-STRUCTURE RESONANCE

BY CONSIDERING NONLINEAR BEHAVIOR

OF THE SITE MATERIALS

D. BRATOSIN1, B.F. APOSTOL2, S.F. BALAN2

1Institute of Solid Mechanics – Romanian Academy, Calea Victoriei 125, Bucharest, Romania,

E-mail: [email protected] 2National Institute for Earth Physics, Călugăreni Street, 12 Măgurele, Romania,

E-mail: [email protected], [email protected]

Received December 22, 2016

Abstract. An appropriate estimation of the site-structure resonance period involves

the nonlinear dependence of the site natural period on strain or loading level. The site

materials are assumed in this paper to be nonlinear viscoelastic materials, modeled by

a nonlinear Kelvin-Voigt model. By using resonant column tests we can quantify the

nonlinear dependence of the site natural period in the normalized form n nT T ,

such that the nonlinear site natural function acquires the form: 0 g nT T T , with

T0 being the fundamental period from seismic recordings. The method consists in

laboratory experiments on soils samples for a site with sufficient seismic records. In

order to mitigate seismic risk, the structures fundamental period must be outside the

range of the computed nonlinear values.

Key words: soil-structure resonance, viscoelastic nonlinear behavior.

1. INTRODUCTION

The strategy for reducing earthquake destructive effects in a country with

significant seismicity like Romania is of uttermost importance. Resonance

phenomenon could lead to buildings significant damage and even to collapse

(including strategic and important cultural and industrial objectives). In this

respect, avoiding resonance is an essential requirement in building design. For this

reason a correct evaluation of the natural (fundamental) frequencies of buildings

and natural (fundamental) frequencies of soils appears to be very useful.

In view of the importance of the subject in mitigation seismic risk in a

country like Romania, [1, 2] authors gathered the most relevant points of their

studies in the domain of site-structure resonance in a single, larger paper, for a

better understanding the subject and its impact on having safe buildings.

Romanian Journal of Physics 62, 808 (2017)

mailto:[email protected]

Article no. 808 D. Bratosin, B.F. Apostol, S.F. Balan 2

In seismological engineering, it is necessary to consider oscillating systems

with nonlinear components of the displacement induced by nonlinear behavior of

the materials of the site exposed to strong seismic action [3, 4, 5]. Resonance

behavior of the nonlinear oscillating systems involves specific points that may

modify linear concept of the natural frequency.

In any dynamic system the coincidence between system natural period and

excitation period causes resonance phenomenon, leading to quite high dynamic

amplifications. This would imply a strong induced seismic stress, which structures

have to bear. The essence of resonance-avoiding strategy consists in an appropriate

knowledge of the two natural periods, one of the structure (TS) and one of the site

(Tg), where TS ≠ Tg. Since nonlinear behavior of site materials is present (stiffness

degradation and increasing of damping), natural period of the site becomes a

function which depends on seismic excitation level, acquiring non-unique value,

and the soil-structure system becomes a nonlinear oscillating system.

In the common practice classical relationship 4 /g sT H v is used, where H is

the depth to the bedrock and vS is the S-waves velocity determined in situ. This

method treats the site as a linear elastic half-space, which is not a satisfactory

approximation for strong displacement (due to high earthquakes magnitudes).

Making use only of one value for the site natural period could lead to an improper

design of the buildings in regions with strong earthquakes.

At the conceptual level the necessity for nonlinear modeling of the behavior

of the soils strata in local sites is known and accepted. The analysis of the impact

of the nonlinearity on the fundamental period is scarce in the literature, the

majority of the bibliographical references being those of the present authors [6, 7].

Studies on resonant nonlinearly oscillating systems are present in the literature of

mechanics, but without connection to the nonlinearly oscillating soil-structure

systems subject to strong earthquakes. There exists a continuous effort for

determining the effects of the seismic amplification induced by the nonlinear

response of the package of crustal layers under the action of strong earthquakes [8].

Numerous scientific presentations exist at internal and international conferences,

related to the so-called in situ effect, which is meant to attempt to solve this

problem. Experimentally, one tries to make a connection between traditional

seismology and the seismological engineering with its today standards, the interest

being on the economical side first, in order to know what happens from the

mechanical-physical standpoint in the close vicinity and in large industrial and

technical buildings (dams, industrial and civil structures and platforms etc.). An

important research direction in nonlinear seismology is given by the observational

data which indicate the influence of the high frequencies in the amplification of the

seismic waves for some locations, although such amplifications have also been

observed for low frequencies (0.05–2 Hz) in the seismic signal propagating through

the poorly consolidated surface layers which exhibit nonlinear behavior for strong

mechanical actions (strong earthquakes) [9].

3 Avoidance strategy for soil-structure resonance Article no. 808

One can also notice more and more research related to the instrumental

seismology in the area of the nonlinear seismology, with the aim of realizing as

safe buildings as possible, with efficient economical expenses. Up to now only a

few scientific publications exist in this field, with a direct practical applicability,

though collateral research has been done in our country at various scientific

institutes, related to local seismic effects. The results of these studies, presented at

international conferences, have attracted a genuine interest from the international

scientific community [2, 7, 10]. We envisage the application of these results, in a

real fashion, to the field of civil engineering and to the isolation of the civil and

industrial buildings exposed to the effects of strong seismic movements.

Considering that geological materials of the site are viscoelastic nonlinear, it

is appropriate to use Kelvin-Voigt model for their behavior. In terms of this

approach the nonlinearity is described through the dependence of mechanical

parameters like shear modulus (G) and damping (ζ) on the strain tensor G = G(γ),

ζ = ζ (γ); thereby all dynamic characteristic of the system depend of the strain

tensor.

It was observed experimentally [11] that for an increasing external strain the

rigidity diminishes, due to degradation effects, while the damping increases. The

shear modulus function decreases and the damping’s tendency is to increase. Such

a behavior affects structural dynamic response and resonance assessment. Under

these simultaneous actions the model considered by authors to evaluate resonance

behavior of the soil-structures (the nonlinear Kelvin-Voigt (NKV) model) appears

to be appropriate, through its capacity to describe nonlinear response of the

materials. Nonlinear amplification functions of the NKV model are therefore

suitable to describe resonance behavior of the system in nonlinear regime [6].

The aim of this study is to find a way to determine nonlinear dependence of

the natural period of the site on strain or loading level, by combining available in

situ data with laboratory computed data. From laboratory tests on resonant columns

the nonlinear dependence of the natural period of the soil samples on strain or

loading is determined, in normalized form, Tn, and from in situ measurements the

initial value T0 is determined; by combining them, the natural period of the site

Tg(•) = T0 ∙ Tn(•) follows. The nonlinear behavior of Tn with strain variations allows

us to estimate some discrete values Tg for a site, that correspond to strong seismic

events (with magnitudes MW > 7.1); this may be useful in avoiding a possible

danger, which may result from placing the fundamental period TS in the determined

range.

2. NONLINEAR NATURAL SITE PERIOD OBTAINED FROM SEISMIC RECORDINGS

Earthquake magnitudes influences maximum accelerations and natural

periods of seismic data recorded in Vrancea seisms making them dependent of it.


This could be seen in Figs. 1–3, where Tg and PGA are directly proportional to

magnitude. Also the maximum predicted event was presented in the figures. The

recordings are from stations located in Bucharest.

A large dispersion of recorded natural periods values happens when using

many stations with different local conditions, but when acquisition is from a single

station the dispersion phenomena meets our requirements (Fig. 3).

Fig. 1 – Nonlinear tendency of site natural

periods.

Fig. 2 – Nonlinear tendency of maximum

accelerations.

In general the density of seismic stations for determining natural period

functions is not satisfactory. INCERC is the only place in Romania where a large

range of seismic movements were recorded from small to the strongest event of

March 4, 1977.

That is why, because of the lack of recorded data, a formula is used,

4 /g sT H v , where H is the site depth and sv is the shear wave velocity; it

provides a unique natural period value, in contrast with earthquake recordings; in

addition, it considers the site as a semi-infinite linear elastic space (Figs. 1 and 3).

Fig. 3 – Seismic records at INCERC site.


3. RESONANCE OF LINEAR SYSTEMS

We consider first a linear system with one degree of freedom subject to a

damped harmonic acceleration,

txtx gg sin0 , (1)

whereo

gx is the amplitude of acceleration (related to peak ground acceleration

PGA) and is circular frequency of excitation. The usual appropriate description

of such behavior is Kelvin-Voigt [12] model (a spring with stiffness k, and a

dashpot with viscosity c connected in parallel, both sustaining a mass m). The

equation of motion is:

txmkxxcxm g sin0 (2)

or:

txxxx g sin2 02

00 , (3)

wherem

k0 is natural pulsation and

m

c

2 is damping ratio.

By using a variable changing t0 and introducing function

0

xtx we get from (3) the dimensionless form for equation of motion

tKC sin''' (4)

the derivatives are taken with respect to τ.

The stationary solution for equation (4) is obtained in the form:

sin,,, , (5)

where , is amplification factor (Fig. 4):

static

dynamic

x

x

,,max, . (6)

As it can be seen from Fig. 4 the dynamic amplification functions have

maximum value at 0 (for small damping), which occurs when the input

frequency is equal or close to the natural period of the system. Also from Fig. 4 we

can see the diminishing of the dynamic amplification effect due to the increasing

damping of the system.


Fig. 4 – Dynamic amplification functions.

4. MODIFICATION OF THE NATURAL PERIOD OF SOILS

USING RESONANT COLUMN TESTS

The modulus-function value iG , damping value i and the corresponding

strain leveli , result from resonant column test under harmonic torsional inputs

with different amplitudes 0 sini iM M t .

A modulus function value iG is obtained from the formula:

2

20

ii i

s

hG v

, (7)

where ρ is the mass density of specimen, i

sv and 0

i are the shear wave velocity

and the sample natural frequency at level i, h is the sample height. In eq. (7) is

the root of torsional frequency equation with analytical form in terms of the ratio R

between torsional inertia of the sample and the torsional inertia of the top cap

system: / topR J J , 2 31 4=

3 45R R R . After performing several tests on

resonant column at different strain level 1, 2...i i n the modulus function


G G and the damping function can be obtained in the normalized

forms:

0

0 ,

n

n

G G G

(8)

where 0G is the initial value of the modulus-function 0 0G G , nG is the

normalized modulus function 0/nG G G ,0 is the initial damping value

0 0 , and n is the normalized damping function 0/n .

Normalized modulus functions (Fig. 5) and damping functions (Fig. 6)

obtained from resonant columns experiments, are presented below. Modulus

functions are decreasing due to dynamical degradation (stiffness reduction) and this

causes an increase in period values.

Fig. 5 – Strenght degradation. Fig. 6 – Damping magnification.

Both strength degradation and damping magnification affect natural period

which becomes dependent of strain level. The natural period of the sample for a

level i is:

0

22 1i

sample i i

hT

G

(9)

and using eq. (8), the nonlinear natural period function of the soil sample follows:

0sample nT T T , (10)

where


0 0

0

0

2 1

1.

sample

sample

n

n

hT T

G

TT

T G

(11)

From the above relationships (11) one may notice that physical and

geometrical properties of the oscillating system are included only in the initial

values (T0) of the nonlinear natural period expression, and normalized expression

(Tn(γ)) depends only on material characteristics. Thus, the resonant column test can

offer accurate data for obtaining only the nonlinear dependence of the normalized

natural period n nT T , while the initial value of the natural period function (T0)

should be obtained from other experiment performed on the nonlinear system or

from real seismic events.

5. DYNAMIC DEGRADATION EFFECT

The material response subjected to torsional loadings is characterized by

nonlinear torsional modulus-function G = G(γ) where the soils are considered

nonlinear viscoelastic materials. Throughout the experiments carried out on our

resonant columns (Hardin and Drnevich), where shear and dynamic torsional

actions are induced, it has been observed that function G values differ from one

loading cycle. Therefore in the test described above, if we keep the strain at a

constant amplitude γ, the stress amplitude decreases, while if we keep the stress at

a constant amplitude τ, the strain amplitude increases [13]. Cyclic loading will

induce in both cases a decreasing in sample stiffness, therefore degradation in

mechanical properties.

A measure for degradation after n cycles is the ratio between modulus

function G of the n-th cycle and the initial value. Then, the degradation function

depends on the number of cycles and on the amplitude, ndd , or ndd , ,

or if we use normalized values with respect to their failure values rr

rr

, ,

so that

0

,,

G

nrGnrd . This degradation function may be determined using test

on dynamic devices by keeping either the strain or the stress under control. An

example is in Fig. 7 where on the triaxial device the stress was maintained for clay

samples. After data processing the expression:


nrbnra

nrd

1

5.0, (12)

is obtained for the degradation function (the diagrams in Fig. 8), where

a(r) = 0.9 + 0.1exp(5.97r), b(r) = 0.45 + 0.05exp(5.73r). The degradation may

increases until the material failure’s if there is an increasing of the loading cycles

and their amplitude. Therefore the failure in dynamic conditions may be defined as

the minimum of degradation function:

0

,minG

Gnrdd

f

f . (13)

Minimum values of degradation function generate in (d, r, n) space a curve r = r(n)

which is obtained by intersection of d = d(r,n) surface with the plane d = df. In

conclusion, the dynamical degradation of site materials of interest induces a

decrease in rigidity, depending on the number and amplitude of the loading cycles.

This behavior can be quantified with the dynamic degradation function or the

dynamic modulus function in normalized form 0/nG G G , both decreasing

with deformation invariant γ. For just one certain cycle n, the degradation function

reduces to the normalized dynamic modulus function

n

r

nctn GrGG

rGnrd f

/

0

, . (14)

Also, it is observed experimentally that a decrease in stiffness is accompanied by

an increase in dissipated internal energy. This phenomenon can be modelled by

means of the damping function, which increases with the deformation level.

Therefore, for the same material (loess), used in the degradation evaluation,

damping function is obtained from resonant columns experiments:

965.7exp134.0153.0 . (15)

Fig. 7 – Clay degradation functions. Fig. 8 – Spatial diagram for degradation

function.


6. RESONANCE OF THE NONLINEAR SITE-STRUCTURE SYSTEMS

The essence of the strategy of avoiding site-structure resonance consists in

the correct evaluation of the natural periods of both the structure sT and the site

gT . However, the site natural period gT has not a unique value. The site materials

have a mechanical behavior strongly dependent on strain, stress or loading level

(expressed by dynamic stiffness degradation and increasing damping) [3, 5, 9, 14].

As a result, it becomes dependent on the earthquakes amplitude, and this

dependence can be observed in the seismic records and can be modelled by the

nonlinear Kelvin-Voigt model.

If we are looking for a qualitative and quantitative evaluation of nonlinear

resonance we consider a single-degree of freedom system subjected to a harmonic

acceleration txtx gg sin0 where o

gx is acceleration amplitude (usually related to

peak ground acceleration, PGA), and is circular frequency of the excitation. In

nonlinear dynamic case we must consider nonlinear dynamic material functions,

; ; G G x x k k x , which now are function of displacement; then the

system is governed by the equation of motion of the nonlinear Kelvin-Voigt model:

. (16)

Using the change of variable t0 and introducing the new time function

0( ) ( ) /x t x we can obtain the dimensionless form of eq. (16), [4]:

sinC K , (17)

where the superscript accent denotes the time derivative with respect to , and:

0

2 2

0 0 0 0

2 ; ; ; g

n static

xc x k xC x K G x x

m m

0

2 2

0 0 0 0

2 ; ; ; g

n static

xc x k xC x K G x x

m m

. (18)

The steady-state solution of the equation (17) can be numerically obtained using a

computer program based on Newmark algorithm [4, 15]. The solution can be

written in the form: , , , sin( ) , where , is the nonlinear

magnification function:

max , ,

,dynamic

static

x

x

(19)

a ratio of maximum dynamic amplitude max dynamicx to static displacementstaticx .


When maximum value of the amplification function is attained, the

resonance frequency of a linear or nonlinear oscillator appears. Formula (19) for

the amplification function of the nonlinear oscillator has multiple resonance

frequencies, which can be obtained by numerical simulation of the behavior of the

specimen in the resonant column, modelled as nonlinear Kelvin-Voigt sample

subject to abutment motion with different acceleration amplitudes

.

By numerical simulations with different values of normalized loading amplitudes

μ we can obtain a set of nonlinear magnification functions .; ct .

Figure 9 presents some magnification functions obtained by using the material

functions nG x and x of a clay specimen tested in the resonant column [3].

For practical applications it is necessary to determine the normalized natural

period in terms of loading amplitude usually described by peak ground acceleration

(PGA). For this conversion – n nT T into n nT T PGA – one can use the same

magnification function (19). Because 0 0/ / 1/ nT T T one can obtain the

magnification functions in terms of normalized period Tn (Fig. 10) and because

a relationship n nT T PGA results (Fig. 11). For

instance, in Figure 12 some functions n nT T PGA for different site materials are

given.

Fig. 9 – Nonlinear magnification functions in terms

of normalized frequency

(for a clay specimen).

Fig. 10 – Nonlinear magnification functions

in terms of normalized periods

(for a clay specimen).


Fig. 11 – Relationship n nT T PGA . Fig. 12 – Some functions n n

T T PGA .

7. EVALUATION OF THE NATURAL PERIOD FOR THE SITE

For the evaluation of the normalized natural periods of the site one must

determine first, by resonant column tests, the nonlinear variation i

nT for each site

stratum, and then one can obtain the average variation of the natural period for the

all site layers av

nT , as the average of the normalized natural period of the strata

weighted with the thickness ih , [6]:

i

i

i

nav

nh

hTT . (20)

For example, we consider seismic station INCERC site stratigraphy (Table 1) and estimate the dynamic modulus function for each material of the site, by using data from resonant columns; then by numerical simulation a function

PGATT i

n

i

n is obtained for each layer. Further, for some PGA values (0.05 g,

0.10 g, 0.15 g, 0.20 g, 0.25 g), using equation (20), mean natural periods for site are computed.

Table 1

Stratigraphy for INCERC site

Material Thickness (m) vs (m/s)

Clay 4.5 460

Sand with gravel 28.5 460

Clays 17 385

Fine sand and clays 17 340

Clays 8 455

Fine sand 53 400


This method, based on resonant columns tests, can be validated by comparison to

data from site recordings. From the spectral analysis of the earthquakes recordings

one may get the dependence of the peak ground accelerations and dominant periods

on magnitude (Table 2). For these data a correlation of type T0 = T0(PGA) is

obtained. The natural periods T0 are increasing functions dependent on seismic

loading, a phenomenon observed from both resonant columns data processing and

seismic recordings (Fig. 13). On this figure it is observed that there are not great

differences between the two acquisition methods.

Table 2

Mean natural periods at INCERC site for different PGA (different seismic events)

Event MW

Recorded

accelerations (cm/s2) PGA (g) T0 (s) Tn

0 0.21 1.00

27.09.2004 4.6 7.97 0.008 0.23 1.07

18.06.2005 4.9 10.53 0.011 0.24 1.12

27.10.2004 6.0 13.69 0.014 0.25 1.16

30.05.1990 6.9 85.20 0.087 0.35 1.63

30.08.1986 7.1 95.72 0.098 0.50 2.33

04.03.1977 7.4 0.250 1.20 5.58

Because strong events from Vrancea seismic source are not produced so

often, data may be used from weak and moderate events and, by extrapolation, we

are able to estimate the behaviour of the natural period for strong events. In Fig. 14

several functions T0 from INCERC site data are presented. It may be seen big

difference regarding periods belonging to the strongest event. This means that the

extrapolation is affected by errors.

In the resonant columns experiments it is possible to induce quite strong

excitations, equivalent to strong seismic deformations. However, the estimation

method of the nonlinear dependence of the strain of the dominant periods by using

resonant columns data implies an interpolation process with much better results,

closest to real cases, successfully used instead of the extension process which

includes uncertainties.

One may conclude that nonlinear variation in normalized form obtained by

this method, together with the determination of the initial normalized value T0

from seismic data, stand for a better approximation of the dominant periods when

strong surface displacement are present (high PGA).


Fig. 13 – Dependence Tn – PGA provided both

resonant column data and seismic records.

Fig. 14 – Partial data extrapolation and

interpolation using resonant columns data.

8. NUMERICAL SIMULATION

In this section we shall use numerical simulation to investigate resonance

behavior of the oscillating systems that underwent degradation effect, by using a

nonlinear Kelvin-Voigt model. For this purpose the material function are those

presented in section 5, i.e. degradation function and damping function:

965.7exp134.0153.0,

1

5.0,

nrb

nranrd . (21)

The initial excitation is considered harmonically damped txg sin with

normalized amplitude values corresponding to strain ones r (0.06, 0.15, 0.30, 0.45),

obtained from resonant columns test, and using the experimental

relationship r045.0 . The damping values ζ (0.05, 0.09, 0.15, 0.20) at these

strain levels are obtained from equation (15) using fr , where γf = 0.01.

From simulations, as depicted in Figs. 15–18, after each cycle, external

loading is applied to the material which have new dynamical properties and

response. As an evidence for nonlinear behavior one can see that peak amplitude of

the nonlinear amplification function depends on strain level r and damping ratio, as

the resonance is encountered at different normalized circular frequencies smaller

than the resonance linear value, showing that the nonlinear behavior of soils

implies an attenuation.

The dependence of the peak amplitude on normalized pulsation is also

observed in the Figures below (resonant peak locus). From these results it is

emphasized both the degradation and the damping effects, during resonance

behavior of the nonlinear oscillating systems. The degradation induces an


increasing tendency of the peak values, as well as frequency shifting (dispersion) at

resonance.

At the same time the damping effect strongly diminishes the peak values.

However, to point out dynamic amplification as an effect of degradation, in Figures

19–22 results of the simulations of the amplification function variation with cycle

number are shown, where strain and damping are maintained constants. While the

degradation induces a dynamic amplification, the damping effect reduces this

tendency. In Figures 23 and 24 the evolution of the peak value of the amplification

function Φmax is presented as an effect of both degradation and damping, in terms

of the normalized circular frequency υ and the natural normalized period

/ 1/nonlinear linear

n g gT T T . In Fig. 24 peak ground acceleration values (PGA) are

used, instead of strain levels ri (generated by them), PGA = 0.458–0.432

exp(–4.32r) from resonant columns tests.

In conclusion, as a result of increasing dynamic loading level, both effects of

degradation and damping lead to a decrease of the peak amplification level and to a

resonance frequency shift corresponding to these peak values. The influence of

damping is, of course, a positive one, as the dynamic amplification decrease. But

the resonance frequency shift may affect the evaluation for a nonlinear system.

The frequency dispersion shown in all figures, prove that oscillating

nonlinear systems have more than one resonance frequency. Straightforwardly,

because of this nonlinear behavior of the site materials, all site-structure systems

have multiple resonance frequencies, depending on dynamic loading levels. For

example, from computations as well as from Fig. 23, considering a strong

earthquakes inducing a PGA = 0.3–0.4g, we may get a 40–60% natural period

shifting.

Fig. 15 – Amplification function

for the second cycle.


for the fifth cycle.



for the 10th cycle.


for the 20th cycle.

Fig. 19 – Amplification function for more cycles

at r = 0.06 and ζ = 0.05.


for more cycles at r = 0.15 and ζ = 0.09.

Fig. 21 – Amplification function for more cycles

at r = 0.3 and ζ = 0.15.


for more cycles at r = 0.45 and ζ = 0.2.


Fig. 23 – Maximum values at resonance

for Φmax= Φmax(υ).

Fig. 24 – Maximum values at resonance

for Φmax= Φmax(Tn).

9. CONCLUDING REMARKS

1) Soil nonlinear systems contain material with dynamic characteristics that

depend on dynamic loading level and the behavior during resonance induces

important changes in their dynamic properties. Site materials have a degradable

rigidity depending on strain level and an increase in the excitation level lead to a

decrease of rigidity and an increase of the natural period values. Therefore, the

rigidity and the natural period are inverse proportional (in both linear and nonlinear

cases).

2) The usual method for determining by calculus the natural period consists

in using “the quarter length formula” ( 4 /g sT H v ), were H is the total depth of

layers and sv is the shear wave velocity. However, this formula considers the site

as a linear elastic space, in contrast with the mechanical reality, and gives a unique

natural period value in contrast with the earthquake recordings.

3) The nonlinear Kelvin-Voigt model seems to be suitable for the evaluation

of the resonant behavior of the nonlinear systems subject simultaneously to

different effects, as rigidity decrease and damping increase, because of its

incorporating two nonlinear functions of material, which include stress modeling

and damping action.

4) Nonlinear amplification functions of the NKV model are suitable for a

quantitative and qualitative description, including the resonant nonlinear regime.

5) While linear oscillating systems have just one resonance period, the

nonlinear ones have several, which depend on the excitation amplitudes. Peak

amplitude values in nonlinear regime arise at different normalized circular

frequencies, which are smaller than the excitation value and smaller than that of the

linear regime, acting as a dispersion phenomenon.


6) Those multiple maximum values of resonance frequencies may lead to

difficulties in the evaluation of the natural period of the site-structure system. For

this reason it appears necessary to apply a modified strategy in the nonlinear

behavior, in order to avoid the resonance phenomenon, for structures exposed to

strong seismic shocks.

7) The resonance strategy avoidance can be compromised by taking into

account only the linear situations which involve only one value of the natural

period.

8) For site dominant period dependency on the excitation level

g gT T PGA seismic data are suitable only if they are cover entire PGA values

domain.

9) The variation of the natural period on excitation level PGATT nn is

obtained using interpolated data from resonant column tests. The method described

here was validated using recorded data from earthquakes.

10) To compensate for the lack of data it is recommended to use resonant

column data for normalized nonlinear variation PGATT nn together with initial

value T0 from seismic recordings, where we approximate the dominant periods of

the site at high PGA.

Acknowledgements. This work has been carried out within the Program Nucleus, supported by

the ANCSI, Romania, projects numbers: PN 16 35 01 04 and PN 16 35 01 07.

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