Lifetime-Oriented Structural Design Concepts- P6 ppt

However, a high-cyclic loading may changethe soil fabric and may lead to an accumulation of permanent deformations.Thus, the serviceability of a foundation is the main concern if it is s

are dissolved This leads directly to an increase in porosity and permeabilityand also to a loss of stability As a rule, no direct deformation of the affectedstructural element is observed in case of a dissolvent process.

2.4.3.1 Sulfate Attack

An external sulfate attack is caused by water and soil layers containing sulfate

or SO2 in the air The sulfate attack can only occur if damp is present Theformation of reaction products (see Subsection 3.1.2.3.3) which cause swelling

in the concrete in sufficient quantities is decisive for a swelling attack followingsulfate penetration The resulting compressive stress due to expansion causesswelling, crack formation and ultimately leads to a loss of stability and damage

to the cement matrix Due to its great technological significance on account

of the prevalence of concrete structures and the sulfate compounds which cur almost everywhere (e.g in ground water, seepage water and soil layers),

oc-a loc-arge number of investigoc-ations into sulphoc-ate oc-attoc-ack hoc-ave been performed

in the past Current knowledge has been integrated into rules and standards[6],[14] No damage has been reported in Germany for concrete with a highsulfate resistance where the measures defined in the standards have been ad-hered to [147] For a number of years there have been international reports of

a new form of sulphate damage to concrete structures; this is described as thethaumasite form of sulphate attack (Subsection 3.1.2.3.3) Unlike the gener-ally known forms of sulfate attack which lead to the formation of cracks andthus to a decrease in stability of the concrete through swelling reactions (et-tringite swelling and, at high sulphate concentrations, also gypsum swelling),

a damaging formation of thaumasite leads to weakening; the strength-formingCSH-phases of the cement matrix are degraded In general, concrete founda-tions of bridges and structures which were exposed to a strong sulfate attack

in the ground are affected (e.g tunnel shells), Figure 2.89 Current knowledge

of the most important damage-relevant factors and the overview of site damage in Germany and abroad are summarised in the progress reportDAfStb7 [147] entitled ”Sulfate attack on concrete” In the [147] special in-terest is taken in the damage potential of pyrite-containing soils in Germany.The oxidation of pyrite-containing minerals (Subsection 3.1.2.3.3) in the ad-jacent stone or soil has been determined in several cases as the cause of thethaumasite form of sulphate attack [291]

thauma-2.4.3.2 Calcium Leaching

If the surface of a concrete structural element is in contact with soft waterover extended periods the calcium hydroxide is broken down hydrolyticallyand calcium in the pore liquid is released (see Subsection 3.1.2.3.2) As a re-sult, the porosity and the permeability of the structure are increased and can

7 German Committee for Reinforced Concrete

Fig 2.89 Concrete damage caused by thaumasite (taken from [151], origin: left

-BRE; right - FA Finger-Institute, Weimar)

Fig 2.90 Corrosion on mortar coatings in two drinking water reservoirs The

coating shown on the right has been almost completely destroyed after about 10years [138]

ultimately lead to a loss of stability The progression of the dissolving and thus

of the damage front takes place very slowly with calcium leaching, especiallyunder environmental conditions which are not constantly damp (several cen-timetres per decade) For normal structures, calcium leaching can generally beclassified as uncritical This environmental attack is, however, significant forstructures which are in direct contact with soft water for an extended period

of time, such as the inside of cooling tower shells and cementitious layers ofdrinking water reservoirs (Figure 2.90) In addition, calcium leaching is a deci-sive damaging mechanism for concrete constructions of nuclear disposal sites,

as the assessment periods for these are several hundred years [804] Furtherstructures for which calcium leaching can be a stability problem are dams,tunnels and water pipes

in the common sense, as it is observed for steel or concrete materials Effectslike abrasion of the soil particles or even fragmentation of the grains are notconsidered here because the design of a foundation usually exclude such states.Furthermore, within the framework of a continuum approach the permanency

of the soil particles is assumed However, a high-cyclic loading may changethe soil fabric and may lead to an accumulation of permanent deformations.Thus, the serviceability of a foundation is the main concern if it is subjected

to a high-cyclic loading In a constitutive relation for soils under high-cyclicloading (see Section 3.3.3) the development of these permanent deformationsmay be modelled similar to a ”fatigue” in steel or concrete materials.Section 2.5.1 discusses possible sources of a high-cyclic loading of soils

It deals with the different appearance of the ”accumulation” phenomenon

in dependence of the boundary conditions (e.g drained or undrained cyclicloading) and outlines the possible consequences for structures

Section 2.5.2 presents a novel definition of an amplitude capturing a tidimensional cyclic excitation The definition is applicable not only to soilsbut also to any other material (e.g steel or concrete) under multiaxial loadingconditions

mul-2.5.1 Settlement Due to Cyclic Loading

Authored by Theodoros Triantafyllidis, Torsten Wichtmann and Andrzej Niemunis

Structures are interacting with the soil The stiffness of the soil depends

on the loading of the foundation and in turn the behaviour of the structure isinfluenced by the stiffness of the subsoil The design of foundations depends

in a great extent on the conditions of the underlain soil and in this way thesoil is forming a part of the building

Uniform settlements of foundations do not produce any structural damage.The admissible settlement may be restrained by serviceability requirementsonly Differential settlements are much more important They may be caused

by local variations of the geotechnical conditions such as a variation of thethickness or the depth of the settlement-sensitive layers, inclusions of softmaterials or non-homogeneities of the void ratio or of the fabric of the soil.Differential settlements may also occur due to different foundation schemes(pile and shallow foundations side by side) and different loadings arising fromthe superstructure design (despite design efforts to avoid this)

traffic loading, e.g high speed or

magnetic leviation trains

watergates tanks, silos surface compaction,

vibro-compaction

power plants off-shore on-shore

Fig 2.91 Sources of cyclic loading of soils

Soil compaction, soil replacement or the choice of a more appropriate

foun-dation design are possible measures prior to the construction to minimize

differential settlements Such procedures have been developed in the past andare not subject of the present study While differential settlements that occur

during the construction process due to unpredictable soil inhomogeneities can

be counteracted to some extent (by ground improvement or a change of the

method of construction), such measures are difficult and expensive during the lifetime of a structure.

With reference to the subsoil, life time oriented design concepts focus onpermanent deformations in the subsoil which occur due to repeated load-ing during the operating time of a structure Examples for such cyclic load-ing caused by traffic (high-speed trains, magnetic leviation trains), industrialsources (crane rails, machine foundations), wind and waves (on-shore and off-shore wind power plants) or repeated filling and emptying processes (water-gates, tanks and silos) are given in Figure 2.91 Furthermore, construction pro-cesses (e.g vibration of sheet piles) and mechanical compaction (e.g vibratorycompaction) introduce cyclic loads into the soil They cause a densification atthe required position which is usually desired for the future construction butmay cause some detrimental effects for the existing neighbours A stress pathdue to a wheel passing on the ground surface is given in Figure 2.92a

In statically indeterminate structures the differential settlements may causechanges of internal forces which may slow down or accelerate the process

of deterioration in the structure Vice versa, a change of the reaction forcesleads to a different rate of settlement accumulation In statically indeterminatestructures under monotonic loading the loading of more compliant foundationsdecreases due to a re-distribution of internal forces The loading of the lesscompliant foundations increases and this may cause plastic deformations in thesubsoil, i.e the settlements of these foundations increase Thus, the differential

Fig 2.92 Cyclic stresses in a soil element a) due to a passing wheel load and b)

due to an earthquake loading

t

ampl av

s

s

Fig 2.93 Accumulation of settlement due to cyclic loading

settlement is reduced For a cyclic loading this smoothing does not always workdue to the decrease of the accumulation rate with the average pressure (Section3.2.2) A life time oriented design concept for structures should include a jointanalysis of the structure and the inhomogeneous subsoil

The settlements (Figure 2.93) due to cyclic loading occur since in an element

of soil closed stress loops, resulting from external loading, lead to not perfectlyclosed strain loops An irreversible deformation remains in the soil, caused byparticle rearrangement due to changes of the intensity and the distribution

of the contact forces between the particles This permanent deformation isaccumulated with the number of cycles Even small amplitudes can signifi-cantly contribute if the number of cycles is high Such a loading with small

amplitudes and large numbers of cycles (N c > 103) is called poly- or cyclic loading As confirmed by the element tests presented in Section 3.2.2

high-and also by parametric studies outlined in Section 4.6.6 the amount of ual settlement depends on the loading of the foundation (average load, loadamplitude) and on the current state of the soil (void ratio, cyclic preloading)

resid-Unfortunately, as demonstrated in Section 4.6.6 differential settlements due

to cyclic loading are much more sensitive (by a factor 3) to inhomogeneities

in the subsoil than those due to monotonic loading

In the context of foundations subjected to cyclic loading, one may

distin-guish between the short-term and the long-term behaviour Studies of the

short-term behaviour deal with the deformation of the structure and the soil within a few cycles (e.g examinations of the dynamic characteristics of asystem) In the majority of such studies a linear response is assumed consid-ering no changes of the soil parameters during the event In the case where anon-linear behaviour of the soil has to be considered an implicit calculationcan be performed as outlined in Section 4.2.11 In long-term studies the ac-cumulation of settlements or changes of the soil-structure interaction are themain concern This book is dedicated to the long-term behaviour

sub-If the load cycles are applied at a low amplitude and low frequency

f = ω/(2π), the inertial forces are negligible and it is spoken of a static cyclic loading If the frequency is large, inertial forces are relevant and the loading is dynamic A harmonic excitation with the displacement

quasi-u = quasi-uamplcos(ωt) can be considered as quasi-static, if uamplω2 is small

com-pared to the acceleration of gravity g Often the amplitude-dependence is ignored and the borderline to dynamic loading is said to lay above f ≈ 5

Hz As reported by the literature and confirmed also by tests of the authors

(with f < 2 Hz and εampl ≤ 10 −3 , [835]) the loading frequency f does not influence the rate of strain accumulation as long as the strain amplitude εampl

is constant

In order to estimate settlements due to cyclic loading and in order to porate them into a life time oriented design concept for engineering structuresone needs special calculation strategies and a constitutive description for thesoil Such a strategy and a high-cycle model have been developed and arepresented in Sections 3.3.3 and 4.2.11

incor-In Section 3.2.2 it is demonstrated for uniaxial cycles with a constant larization that having packages of cycles with different amplitudes their se-quence does not play a significant role for the final value of the permanentdeformation It is further assumed that a transient or periodic signal can bedecomposed into a series of cyclic signals with different frequencies (Section2.5.2) Afterwards these signals are grouped into packages in which the ampli-tude is constant (Figure 2.94) The analysis of the permanent soil deformationcan then be performed as given in Sections 3.3.3 and 4.2.11

po-If the cyclic stresses in the soil are not too close to the failure criterion

and if the amplitudes are below εampl ≈ 10 −5 the accumulation rate can be

expected to become very small or even vanish after a sufficiently large number

of cycles (adaptation, ”shakedown”) Having reached such asymptotic statethe soil behaviour is almost linear elastic during the subsequent cycles Insuch cases accumulation effects need not to be considered in the design ofstructures In Section 3.2.2 it is demonstrated that polarization changes lead

to a temporary increased accumulation rate Having reached an asymptotic

Fig 2.94 Decomposition of a signal with varying amplitudes into packages of cycles

with constant amplitude

state a re-start of the accumulation and adaptation process may occur after

a sudden change of the polarization However, no sound experimental studiesexist on the accumulation at such small amplitudes Thus, the effect cannot

be validated or quantified yet

Another asymptotic state may be observed in saturated cohesive soils porting a foundation which are subjected to a cyclic loading The accumula-tion of pore water pressure (see remarks below) is very small, if the excitation

sup-frequency f is below the ratio c v /b2 with c v being the coefficient of

consoli-dation and b the width of the founconsoli-dation (almost drained conditions) and if the strain amplitude is below εampl≤ 10 −2 [329].

If the cyclic stress path repeatedly reaches the failure criterion an mental soil collapse may occur An application of cyclic loading with smalleramplitudes after a strong event (e.g a storm in the case of offshore founda-tions) can lead at least hypothetically to a ”healing effect”, i.e to a reduction

incre-of deformations imposed by the strong event

A cyclic loading may not only cause permanent deformations Depending

on the boundary conditions it may also result in a change of the averagestress In water-saturated soils under partly drained or undrained conditions

the pore water pressure uavmay accumulate with the number of cycles due to

the contracting soil behaviour Thus, the effective mean pressure pav, the shearstrength and the stiffness decrease or even vanish (so-called ”liquefaction” or

”cyclic mobility” in case of temporary loss of shear strength)

Such effects are observed e.g during earthquakes (Figure 2.92b) While a

”man-made” high-cyclic loading on structures is associated with small tudes and a high number of cycles the number of cycles is small in the case of

ampli-a seismic loampli-ading but the ampli-amplitudes ampli-are lampli-arge The drampli-ainampli-age conditions plampli-ay ampli-asignificant role Usually undrained conditions are considered for an earthquakeloading because of the great intensity and the short duration of action In con-trast, a high-cyclic loading is calculated assuming drained conditions because

of the long duration and the small intensity of action In the undrained case apore water pressure accumulation takes place and as a consequence effects likeliquefaction, phase or layer separation and spontaneous densification (duringre-consolidation) may be observed

These effects can be utilized for an intelligent foundation design in order toestablish a passive screening, i.e to reduce the seismic loading acting on thestructure and thus to prevent it from damage during an earthquake A popularexample for a passive screening are the foundations of the Higashi temple inKyoto In the case of an earthquake layers of fine grained material are brought

to liquefaction in order to avoid the passage of shear waves to overlain layers

or structures (so called ”Hanchiku-effect”) The liquefaction phenomenon isalso utilized for soil improvement techniques (deep vibratory compaction).However, if the described phenomena under an undrained cyclic loading arenot well understood by the design engineer a non-appropriate design of thefoundation may be chosen A more detailed discussion of the effect of a cyclicloading under various boundary conditions is given in Section 3.1.3

Another source of cyclic loading of soil, which is not discussed in detail

in the present book, is caused by climatic changes and seasonal effects Suchloading is connected with changes of the portions of the three phases (solidparticles, pore water, air) of a soil and may lead to changes of its fabric and itsmechanical properties The cyclic change of the water table e.g leads to an ac-cumulation of water content (degree of saturation) in the transition zone and

an alteration of the effective stress and the suction This cyclic change of theeffective stress acting on the solid phase may cause permanent deformations

In the case of cohesive soils permanent deformations are generally associatedwith wetting and drying processes leading to swelling and shrinkage Clusters

of tension cracks may occur influencing the hydraulic and mechanical erties of the soil Such kind of cyclic loading referring to hydro-mechanicalcoupling and partial saturation of soils is of great importance for water reser-voirs, dam embankments, dykes, etc

prop-Sources and effects of cyclic loading are maningfold In a life time orienteddesign all relevant influences and boundary conditions a soil may be exposed

to (depending in turn on the design solution) have to be kept in mind

2.5.2 Multidimensional Amplitude for Soils under Cyclic Loading

Authored by Andrzej Niemunis, Torsten Wichtmann and Theodoros Triantafyllidis

A cycle is understood as a path (a trajectory parametrized by time) which

is recurrently passed through by a state variable (like strain or stress) For

a scalar or tensorial variable  we may define its average value av to bethe centre of the smallest (hyper)sphere that encompasses all states upon

the cycle For a scalar variable one obtains av = 1

2(max+min) and theamplitude isampl= max|−av| For tensorial variables, apart from the size

of the (hyper)sphere, we want to convey some information on the polarizationand the ovality of the path, which renders the amplitude to become a tensor.Further we consider strain cyclesε(t) only, with ε= lnU whereU is the right

stretch tensor We distinguish between in-phase (=IP) strain cycles

-1.0 1.0

1.0

-1.0

0.5 0.5

c) OOP - cycles:

b) multiaxial IP - cycles a) uniaxial IP - cycles

0.5 0.5

-0.5 -0.5

ε 3

ε 1

-0.5 -0.5

for which the variability of all components in time can be described by a

com-mon function f (t) and out-of-phase (=OOP) cycles which cannot be expressed

in this way, e.g

ε=εav+ diag(εampl

11 sin(ωt + ϕ11), εampl

22 sin(ωt + ϕ22), 0) ϕ11 = ϕ22

(2.77)

and which require individual time tracking f ij (t) of various ε ij components

The collection εampl

ij of the amplitudes of the individual components in (2.76)should not be mixed up with the tensorial definition of the amplitude Aε

which will be proposed further The IP-cycles that have only one non-zeroeigenvalue ofεampl are termed uniaxial,

of the strain pathε(t) solving σ(t) =EEE : ε(t) for the unknown ε ij (t) using

the secant elastic stiffnessEEE of the cycle Similarly, given the cumulative ratesfrom experiments (measured (= m) or prescribed), namely pseudo-relaxation

˙

σm(N c) and pseudo-creepDm(N c), the constitutive strain accumulation rate

Dacc is obtained by solving the material equation ˙σm =EEEm : (Dm− Dacc).Note that all rates are meant as increments or residuals after a single cycle inthe high-cyclic context

A description of polarization must involve all 6 components of the strainpathε(t) because the strain states need not be coaxial upon a cycle In order to

evaluate the tensorial strain amplitudeAεfrom a discrete pathε(t1), ε(t2),

obtained from laboratory tests or from FE-calculations one should avoid usingthe first cycle (= irregular cycle discussed in Section 4.2.11) From a represen-tative (recorded or calculated) cycleε(t) we extract the resilient strain path

ε e (t) It is done by subtracting the residual (cumulative) portion creep) from it This operation is called detrending The proposed detrending

(pseudo-procedure consists of four steps:

• Calculate the hodograph D(t) ≈ ˙ ε(t), Fig 2.96

2 2

Dacc

Dacc D

1D

D

2Da)

b)

Fig 2.96 A hodograph is a trajectory ofD (t) ≈ ˙ ε (t) parametrized with time t,

analogously to the strain pathε (t) The rate of accumulation can be easily identified

as a drift rate (denoted with arrow) of the average strain upon a cycle Note that thestrain rate is an exactly periodic functionD (t) = D (t + N T ) whereas the strain ε (t)

is not The distinction between a) the cycles encompassing some area (out-of-phasecycles (= OOP) and b) the open-curve cycles is of importance

Fig 2.97 Determination of spansR(3)withr(3), R(2)withr(2)and R(1)withr(1)

for a 3-d loop

In the following sections the index eis omitted for brevity

A tensorial definition of the strain amplitudeAε has been proposed [575]

to consider the observations (Section 3.2.2) that apart from the size of thestrain cycle also its ovality, deplanation and the changes of its polarization canstrongly influence the rate of accumulationDaccwhich is the most importantelement of the high-cycle model (Section 3.3.3) The proposed definition hasproven to work well with various convex strain cycles similar to harmonicoscillations and to consider the experimental observation that the change of

circulation of circular cycles does not affect the rate of accumulation.

Suppose, we are given a detrended strain cycle in form of a sequence ofdiscrete strainsε(t k ), k = 1, , M recorded (smartly to save the computer

memory) by an FE program at a Gauss integration point Fromε(t k) with

k = 1, , M we determine the pair of the two most distant states, say ε(t a)and ε(t b ) The so-called span of the cycle is quantified by its size 2R(6) =

ε(t a)− ε(t b ), wherein − →

denotes normalization The upper index (i)corresponds to the dimensionality

of the strain cycle For example the original strain path (before so-calledflattening, see below) can be at most six-dimensional,ε(6)(t) = ε(t) In order

to find the second longest span the original strain path is projected onto the

hyperplane perpendicular to r(6) The resulting flattened strain trajectory

ε(5) =ε(6)− r(6) : ε(6)⊗ r(6) has at most five dimensions The span of the

flattened trajectory can be determined analogously and described by R(5)and r(5) The flattened loop is subjected to the subsequent projection, thistime alongr(5), etc Of course R(6) ≥ R(5)≥ · · · ≥ R(1) holds The tensorialamplitudeAεis proposed to be the following sum of dyadic products

collecting all spans Summing up, the method consists in the stepwise tion of spans and the degeneration (flattening) of the strain path The sense ofthe direction ofr (i) is of no importance, which is obvious from (2.80) Projec-tions from a 3-dimensional path to the 1-dimensional path are shown in Figure2.97 For the 4-th order tensorAε the list of radii R(6) ≥ R(5) ≥ · · · ≥ R(1)can be seen as the spectrum of eigenvalues and the mutually orthonormal ori-entationsr(6), r(5), , r(1) are the eigentensors The normalized amplitude

in Section 3.2.2

If several sources of cyclic loading are acting simultaneously, complex strainloops may result from different polarizations and frequencies of the cycles Aprocedure for the determination of the strain amplitude and the number ofcycles is discussed in the following Some problems related to the definition

of the amplitude are:

• After a full cycle the strain path does not exactly pass through the same

strain state (due to accumulation) Moreover the strain loop may intersectitself which does not indicate that the loop is over It is evident that amathematical tool is required in order to detect the period, i.e when astrain loop is finished

• Suppose a strain loop has been prescribed by two spans with slightly

differ-ent frequencies so that a slow rotation of polarization occurs, Fig 2.98a,b

If two spans were equally polarized the beat would occur The hithertohypothesis either ignores the small spans or overestimates their effect de-scribing such loading as distinct packages with alternating polarization

• It is not clear if smaller but faster cycles in plane with the dominant cycle

or out of this plane may be ignored, Fig 2.98c,d

For practical applications in soil mechanics the detrended strain path ε ij (t)

can be assumed to be a superposition of individual harmonic oscillations The

harmonic oscillations can be distinguished judging by the frequency f K (or

angular velocity ω K = 2πf K ) From each of six components ε ij (t) of the

-0.5 0 0.5 1

-0.2 0 0.2

-1 -0.5 0 0.5

-1 -0.5 0 0.5

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5

-1 -0.5 0 0.5

Fig 2.98 Strain paths (Lissajous curves) obtained from the superposition of sine

functions a),b) with slightly different frequencies and amplitudes or c),d) with verydifferent frequencies and different amplitudes e) Acceleration measured on groundsurface during the Niigata Earthquake (1964)

strain path we pick up a portion which corresponds to a common dominant

frequency f K The componentwise sum of these six signals constitutes a monic oscillation In general it is a 6-dimensional ellipse in the strain space.

har-In this Section the oscillations are numbered with the capital letter K We will try to approximate the signal ε ij (t) as a sum of M oscillations:

The essential purpose of the present spectral analysis is filtering out the

por-tions of the individual strain components ε ij corresponding to the same

angu-lar velocity ω K and gathering them into common oscillations This needs to

be done only for several dominant frequencies f K with K = 1, 2, for which the strain amplitudes εamplK

ij are large Since the square of the amplitude tates the accumulation rate (Section 3.2.2) the impact of smaller amplitudesbecomes negligible (assuming the superposition of their effects)

dic-Each component function ε ij (t) is treated as a series of discrete values ε ij(k) given at k = 0, 1, , N − 1 (N is an even number) points equally distributed along the time axis over the time window from t = 0 to t = (N − 1)Δ Denot- ing the sampling interval as Δ we find the Nyquist frequency f c = 1/(2Δ) and the (complex valued) discrete Fourier transform (DFT) Y ij(n) of the discrete

strain path ε ij(k) with frequencies f n = ω n /(2π) such that −f c < f n < f c.

The frequencies are indexed with n = −N/2, , −1, 0, 1, N/2 and

capture all strain oscillations of importance and that no higher frequenciescan leak into the (−f c , f c) range (no aliasing) Plotting the tensorial norm(sum overij only) we obtain the periodogram

Y (n)=

⎧

⎪

⎪

|Y | ij(n) |Y | ij(n)+|Y | ij(−n) |Y | ij(−n) for n = 1, 2 , N/2 − 1

(2.85)

and among the frequencies f n = n/(N Δ) we may find the one for which the (real valued) function Y (n) has its maximum This frequency is denoted as f K and the corresponding angular velocity ω K = 2πf K enters (2.83) Technically,

since the original signal ε ij (t) may be a pure sine function with a frequency lying in the middle between two adjacent f n-s one needs data windowing(apodization, e.g Hann or Barlett window) in order to reduce the leakage

of frequency Having found the dominant frequency f K we filter out a bandaround this frequency from each component of the strain For this purpose

we simply multiply each DFT Y ij (f n ) by the band-pass filter H K

BP (an evenfunction equal to unity in the vicinity of±f K and to zero elsewhere) in the

frequency domain We obtain six f K-band-pass filtered transforms

correspond to the double amplitudes 2εamplK

ij of the oscillation K and enter (2.83) The expression for DIFT contains an n-sum from 0 to N (instead of

from −N/2 to N/2) thanks to the N-periodicity of the DFT, i.e Y ij(−n) =

Y ij(N −n) The phase shift ϕ K

ij is calculated from the correlation of individualcomponentsij For example, we may assume ϕ K

11= 0 and the phase shift ϕ K

22

is calculated in the frequency domain using the product Y K

11 Y ∗K

22 whereinthe asterisk denotes the complex conjugation The phase shift follows from

the time lag τ22 obtained as the time shift for which the DIFT of the above

product has its maximum Finally we have ϕ K

22= ω K τ22 The K-th oscillation

is completely determined by repeating analogous calculations of correlation for

all strain components ε ij The remaining oscillations are selected analogously

using the reduced signal

center the fatigue loading independently

A single load package of duration T is calculated as T

K f K

εamplK2

i.e.without considering the mutual polarizations of different oscillations withinthe package

An experimental proof of the proposed analysis will be given in future