Recent advances in the stochastic modelling of the seismic ... · PDF fileRecent advances in...

Recent advances in the stochastic modelling of the seismic action and evaluation of the structural response

Pierfrancesco Cacciola

Senior Lecturer in Civil Engineering ( Structural Design )School of Environment and Technology, University of Brighton, Cockcroft building, Lewes Road, BN2 4GJ, Brighton, UK

• Deterministic and Stochastic vibrations of linear and non-linear systems

• Identification and health monitoring of vibrating structures

• Computational Mechanics and Re-analysis techniques

• Earthquake Engineering

Research Topics

- Stochastic modelling of the seismic action: spectrum compatible approach

- Stationary and nonstationary stochastic response

Randomness in EarthquakeEngineering

• When an earthquake occurs, the resulting ground motion produces in structures internal stresses and strains varying with respect to time.

• If the ground motion time history is known and if we are able to accuratelymodel the structure the dynamic response can be straightforwardlydetermined.

• Perhaps no other discipline within engineering has to deal as much uncertainty as the field of earthquake engineering

• The randomness in the occurrence of earthquakes in time and space, the vast uncertainty in predicting intensity/time-history of the resulting groundmotions, and the inhability to accurately assess capacities of structuresunder cyclic loading all compel us to make use of probabilistic methods in order to consistently account for the underlying uncertainties and makequantitative assesment of structural safety.

• The definition of reliable models of the seismic action is mandatory.

Modeling the seismic action

- Probably the right model should be defined following the phenomenology of earthquake ground motion (fault mechanism, wave propagation in nonlinear multiphase unknown medium, … too many unknowns)

- Definition of probabilistic models based on free field (or relatively small depth) ground motions (response spectra, records of ground motion time-histories).

As much information we have as much the model is rich: seismic codes provide the response spectra for each site, database of earthquakes (time histories, synthetic data).

Spectrum compatible representation of the seismic action

( )paS T( ) ( )igu t&&

( ) ( )jgu t&&

• Problem position: given a response spectrum determine ground motion time-histories

( )( )

( ) ( )( )

pa paj

S T S Tn =

= ∑% %

Target and SimulatedResponse Spectra

Target Response Spectrum(i.e. Eurocode 8)

• Inverse problem: the solution is not unique !- Deterministic- Probabilistic

Stationary/Quasi-Stationary Stochastic Approach

Target Response Spectrum(i.e. Eurocode 8)

( )paS T

( ) ( )igu t&&

( ) ( )jgu t&&

( )guG ω&&

Spectrum-Compatible Power Spectral Density : zero-meanGaussian stationary process

• a Gaussian stochastic process is fully defined by the knowledge of the Power Spectral Density function

• Due to the random nature of the seismic action the most suitable approach for generating spectrum compatilbe accelerograms is based on theory of the Gaussianstochastic process.

Stationary/Quasi-Stationary Stochastic Approach

• Assuming the ground-acceleration process as zero-mean Gaussian stationary process the pseudo-acceleration response spectrum can be approximated by the expression (Vanmarcke and Gasparini, 1977)

20 0 0 , 0,( , ) ( , 0.5; )pa U s i U US T pω ζ = ω η = λ λ

, ; 0,1, 2i U iλ =

( ) ( )g

iui U GH d

λ = ω ω ω ω∫ &&

2 2 2 2 2 20 0 0

1( )( ) 4

H ω =ω − ω + ζ ω ω

are the response spectral moments

is the time observing window is the not-exceeding probability

is the peak factor

• That has been commonly used in literature as a vehicle for determining the spectrum-compatible power spectral density of the ground acceleration process(see e.g. Vanmarcke and Gasparini, 1977; Kaul, 1978; Sundararajan, 1980; Pfaffinger, 1983; Preumont, 1983; Spanos 1985; Preumont, 1985; Di Paola and La Mendola, 1992; Deodatis 1996; Falsone and Neri, 2000; Cacciola, Colajanni and Muscolino, 2004).

( )guG ω&&

P. Cacciola, P. Colajanni, G. Muscolino, Combination of Modal Responses Consistent with Seismic Input Representation, Journal of Structural Engineering (ASCE), 130(1), pp. 47-55, 2004

• First direct spectrum-compatible PSD

210 1 0

( , )4( ) ( )4 ( , )g g

u i u jji i U i

ω ζζω = − ∆ω ω ω π − ζ ω η ω ζ

∑&& &&

with η = ( ) 1 22ln 2 1 ln 2.U U U UN exp N − −δ π

02 20 0

1 21 1 arctan1 1

ζ δ = − − − ζ π − ζ

( ) 1ln2

TN p −= ω −π

• After determining the spectrum-compatible power spectral density the simulation of stationary or quasi-stationary spectrum-compatible ground acceleration processes is pursued via the equation (see e.g. Shinozuka and Deodatis, 1991)

( ) ( )

1( ) 2 ( ) cos( )

g u ii

u t G i i tϕ ω ω ω φ=

= ∆ ∆ ∆ +∑ &&&&modulating function

random phase

GASMA Code (Visual Fortran)

0 20 40 60 80 100

0.003( )

guG ω&&

0 1 2 3 40

0 10 20 30-1

( )paS T

• It could be integrated in various FE code (Code-Aster ?)

• The proposed spectrum compatible PSD allows to overcome the limits Complete Quadratic Combination (CQC) rule used nowadays by practitioners avoiding the above approximations

• The Complete Quadratic Combination (CQC) rule has been derived via a stochastic approach (Der Kiureghian, 1980 ) assuming that: 1) the ratio between nodal and modal peak factors , are approximately equal to one; 2) white noise ground motion process

eηjdη

P. Cacciola, P. Colajanni, G. Muscolino, Combination of Modal Responses Consistent with Seismic Input Representation, Journal of Structural Engineering (ASCE), 130(1), pp. 47-55, 2004

• Very often practitioners are not aware that they use stochastic approach in the response spectrum based seismic design

2 21 1

( , ) ( , )max ( )

m mpa j j pa k k

j k jkj k j k

S Se t

ω ζ ω ζ= ψ ψ ρ

ω ω∑∑(CQC rule)

2 21 1

( , ) ( , )max ( )

m mpa j j pa k k

j kj k j

e jkd dk

S Se t

η ρη

ω ζη

ω ζ= ψ ψ

ω ω∑∑

Two examples in which CQC rule fails

Eccentric high rise building(slender structure)

Rigid structure with an appendage

• In both cases CQC provide about 10-20% underestimation of maximum displacement• The proposed approach reduces the error to 1-3%• US Nuclear regulatory have already included in 2001 approximate correlation coefficients

Nonstationary Stochastic Approach

• While stationary/quasi stationary approach provides accurate results for most linear behaving structures, on the other hand the importance of non-stationary frequency content on the response of nonlinear structures has been manifested in various studies (see e.g. Yeh and Wen, 1989; Wang et al. 2002). Thus, more reliable simulations of earthquakes have to take into account the time variability of both intensity and frequency content of the ground motion.

• Few contributions for the non stationary case: Spanos and Vargas Loli(1985), Preumont (1985), Deodatis (1996), Shrikhande and Gupta (1996), Lin and Ghaboussi (2001) , Gupta and Mukherjee (2002)

• It has to be emphasized that when one deals with nonstationarity a certain degree of subjectivity is introduced. Nonstationary behaviour of earthquakes depends by several factors (rupture mechanism, path of the seismic wave, local soil condition)…. Further information: database of earthquake

P. Cacciola, A method for generating fully nonstationary spectrum-compatible earthquakes, Fifth International Conference on Computational Stochastic Mechanics, 21-23/06/2006, Rhodes (Rodos), Greece

corrective term: zero-mean Gaussian stationary process whose power spectral density have to be determined

• Proposed spectrum compatible ground motion modelfully nonstationary recorded/simulated signal

) ( ))( (R Sg g gu t tt uu= + &&& &&&

• Approximating the target response spectrum by the following equation2 2( ) ( )R S

pa pa paS S S≅ +

• using the procedure proposed by Cacciola, Colajanni and Muscolino (2004)

( )22 10 002

10 1 0

4( ) ( )4 ( , )g g

R ipa i pa iS S

u i u jji i iU

G G−

ω ζ − ω ζζ ω = − ∆ω ω ω π − ζ ω η ω ζ

∑&& &&

( ) ( )

1( ) ( ) 2 ( ) cos( )

Nr R S r

g g u ii

u u t t G i i tϕ ω ω ω φ=

= + ∆ ∆ ∆ +∑ &&&& &&

• after determining the power spectral density pertinent to the corrective process, the simulation of fully nonstationary spectrum-compatible ground acceleration earthquakes is pursued via the equation

( , ) ( , )S S

Numerical Results

0 10 20 300

( )Rgu t

[ ]t s

0.95)a

( )RSA Tcms

0 1 2 3 40

1200target (EC8)original2

[ ]T s

-500-250

0250500

-500-250

0250500

0 10 20 30-500-250

0250500

( )gu t

[ ]t s

( )gu t

0 20 40 60 80 1000

[ ]/rad sω

Numerical Results

( )RSA Tcms

[ ]T s0 1 2 3 4

1200target (EC8)originalmodified

P. Cacciola, A stochastic approach for generating spectrum compatible fully nonstationary earthquakes, submitted Computers and Structures, 2008

0 10 20 300

30000originalmodified

[ ]t s

( )E t

• Instantaneous energy

( ) ( ) ( )R SE t E t E t= +

0 10 20 300

originalmodified

[ ]t s

• Instantaneous frequency

( ) ( ) ( )( )

( ) ( )

R R S S

t E t E tt

E t E t

ω + ωω =

Evolutionary spectrum compatible power spectral density function

( ) ( , ( ) ( , )( ( ,)) ) g g g

R Rg uu

Sg uu u tt G tu t t tG G ωω ω= + ⇒ = +&&&& &&& && &&&

Recorded accelerogram Quasi-stationary term: zero-mean Gaussian process whose PSD has to be determined

Evolutionary PSD associated to the recorded accelerogram (Peng & Conte, 1997)

( , ) ( ) ( );g

NRu k k

G t a t G=

ω = ω∑&&

2 2 2 2

( ) ( ) ( )

1 1( )2 ( ) ( )

k k ktk k k k

k k k k

a t t e U t

β −γ −ζ = α − ζ − ζ

νω = + π ν + ω+ η ν + ω− η

joint time-frequency distribution of the PSD of the recorded accelerogram

- parameters determined in order to approximate in a least square sense the power spectrum of the recorded accelerogram.

- Actually, the parameters can be determined also for fitting various data from a database

, , , , , ,k k k k k kN α β γ ν η ζ

Evolutionary spectrum compatible power spectral density function

[ ]/rad sω[ ]t s

( , )guG t

[ ]/rad sω[ ]t s

( , )guG t

&& )b( , )g

RuG tω&&

( , )guG tω&&

Time-frequency distribution of El Centro 1940 recorded accelerogram

Evolutionary Spectrum compatible PSD

- The evaluation of the spectrum compatible EPSD is independent of the model used for G t( , )g

Ru ω&&

Ground motion spatial variability

• Ground motion arising from seismic waves is a phenomenon that for its nature varies with time and space.

• Earthquake ground motion spatial variability can influence significantly the response of structures, especially if large or rigid. In this context, a number of contributions has been devoted on the study of the effects of spatial variability on various structures including buildings, bridges, arcs, dams, rigid foundations, pipelines and transmission lines.

• In order to take into account for spatial variability, ground motion should be modelled by stochastic vector process

• Gaussian vector processes are fully defined by power spectral density matrix

Problem Position: multi-variate case

Point 3

Point 2

Point 1

40 m30 m

0 1 2 3 40

(2) ( )RSA T

(3) ( )RSA T (1) ( )RSA T

( ) ( )igu t&&

( ) ( )jgu t&&

( ) ( )igu t&&

( ) ( )jgu t&&

( ) ( )igu t&&

( ) ( )jgu t&&

•Spectrum compatible

•Multi-correlated

M-variate nonstationary ground motion processes

•Spectral-based-representation approach (Shinozuka and Deodatis, 1988)

0 ( ) 0, ( 1,..., )jE f t j m = = Zero mean

0 0 011 12 10 0 0

0 21 22 2

0 0 01 2

( , ) ( , ) ( , )( , ) ( , ) ( , )

( , ) ( , ) ( , )

m m mm

S t S t S tS t S t S t

S t S t S t

ω ω ωω ω ω

ω ω ω

M M O M

Evolutionary Power spectral density matrix

( , ) ( , ) ( ), 1, 2,...,

( , ) ( , ) ( , ) ( ) ( ) ( ), , 1, 2,..., ;

jj j j

jk j k j k jk

S t A t S j m

S t A t A t S S j k m j k

ω ω ω

ω ω ω ω ω ω

= Γ = ≠

Coherence function

Fully nonstationary: non separable function

Simulation Formula

• If the evolutionary power spectral density matrix is known samples of ground motion time-histories can be simulated via the following procedure (Deodatis, 1996)

• Cholesky’s decomposition

• Simulation Formula

0 *( , ) ( , ) ( , )Tf t t tω ω ω=S H H

( ) 2 ( , ) cos ( , ) , 1, 2,...,m N

j jr s s jr s rsr s

f t H t t t j mω ω ω ϑ ω φ= =

= ∆ − + = ∑∑

1Im ( , )

( , ) tanRe ( , )

ωϑ ω

random phases

P. Cacciola and G. Deodatis , A Method for generating fully nonstationaryspectrum compatible ground motion vector processes, Meccanica Stocastica ’08 Cefalù, June, 11-12th, 2008

• It is assumed that the nonstationary spectrum compatible ground motion vector process is given by the superposition of two contributions:

- a fully nonstationary vector process, with known nonseparable power spectral density matrix representative of geological and seismological conditions of the site in which theground motion have to be simulated;

- quasi-stationary vector process, with unknown power spectral density matrix, aimed to correct each component of inorder to make them spectrum compatible

( ) ( ) ( ), ( 1,..., )SC L Cj j jf t f t f t j m= + =

•Approximate response spectrum: set equal to jth target one2 2

( ) ( ) ( )( ) ( ) ( )SC L Cj j jf f fRSA RSA RSAω ω ω = +

unknownknown quantities

•Response spectrum of the corrective term

( ) 2 ( ) ( )0 0 0 0 00,

( ) ( , ; , 0.5) ( )Cj

C Cf j j

sU URSA T pω ω η ω ζ λ ω= =

2( )0 0,

( )( ) ( , ) ( )cCjj i

i U fGH d

ωλ ω = ω ω ω ω∫

unknown Power spectral density of the corrective term

•After simple algebra( )( ) ( )22 2 4 ( ) ( )

0 0 0 0 00,( ( )) ( ( )) ( ) ( )j jj j

L LC C

(f ) (f )j jU URSA RSA RSA RSAω ω ω η λ ω ω− = ∀ >

•Power spectral density of the corrective term (Cacciola, 2006)

( ) ( )( )

( ) ( )10 0( ) ( )0

2( ) 10 1 0

( , ) ( , )4( ) ( ) ,4 ( , )

fii ij j

i rf fj ri i iU

RSA RSAG G

ω ζ − ω ζ ζω = − ∆ω ω ω π − ζ ω η ω ζ

∑22 L

• Iterative improvement•Evolutionary Spectrum compatible power spectral density matrix

( ) ( )( , ) ( , ) ( , ),

( , ) ( , ) ( , ) ( ),

L CSC j jjj f f

SC SC SCjk jj kk jk

S t S t S t

ω ω ω

ω ω ω ω

( )( )

( )( , ) ( )

GS t t

ωω φ=

Modulating function

Numerical application

•Three different locations in which fully nonstationary ground motion samples compatible with three different response spectra defined by Eurocode 8 have to be simulated.

Point3

Point 2

Point 1

40 m30 m

0 1 2 3 40

(2) ( )RSA T

(3) ( )RSA T (1) ( )RSA T

type A soil

type B soil

type D soil

( ) ( )

1 1.5 0

( 1, 2,3)

jg B C

j Cg C D

j C Dg D

TRSA T a S T TT

RSA T a S T T T

TRSA T a S T T TT

T TRSA T a S T T sT

= + ≤ ≤

= ≤ ≤

0.35ag g=

Numerical application 11 12 13

21 22 23

31 32 33

( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , )

L L L L

S t S t S tt S t S t S t

S t S t S t

ω ω ωω ω ω ω

ω ω ω

0 1 2 3 40

(3) ( )RSA T

(2) ( )RSA T(1) ( )RSA T

( ) ( 1, 2,3)j

L(f )RSA T j =

( )RSA Tmms

( )jkγ ω

0 20 40 60 80 1000

23( )γ ω( )γ ω

13( )γ ω

( , )LjjS tω

20 2 2

1 4 ( )( ) ( )

( , ) ( ) ( )

1 4 ( ) 1 4 ( )( ) ( ) ( ) ( )

g fg g f f

S t A t S t

t tt t t t

ω ωζω ω

ωω ω ω ωζ ζ

ω ω ω ω

+ = − + − +

2 2( ) exp (1 ) (1 )exp (1 )

( ) ( )jk jk

jk a a a a a aξ ξ

γ ω α ααθ ω θ ω

= − − + + − − − +

•Local EPSD matrix•Comparison between local response spectra and target ones

122 2 2 2

( ) ( )( )

( ) 22 ( )10 0( ) ( )0

2( ) 10 1 0

( , ) ( , )4( ) ( ) ,4 ( , )

fii ij j

i rf fj ri i iU

RSA RSAG G

ω ζ − ω ζ ζω = − ∆ω ω ω π − ζ ω η ω ζ

•Direct power spectral density functions

( ) ( )( , ) ( , ) ( , ), 1, 2,...,L CSC j jjj f f

S t S t S t j mω ω ω= + =

11 ( , )SCS tω

22 ( , )SCS tω

33 ( , )SCS tω

(3) ( )RSA T(2) ( )RSA T

(1) ( )RSA T

( )RSA Tmms

0 1 2 3 40

1500target (EC8)10 samples1000 samples

•Response spectra•Samples at three different locations

0 20 40 60 800

1( )jkγ ω 23( )γ ω

12 ( )γ ω

13( )γ ω

•Coherence functions

0 10 20 30-600

1 ( )SCf t

( )SCf tmms

If we do not have have therecords ?

If the records are very far fromthe site ?

Coherence function ?

Site response: wave propagation

( ), ( )z G zρ

Bedrock

Ground surface

( , ) ( , )( )z t u z tzz t

∂τ ∂= ρ

∂ ∂- Linear approach: frequency domain

- Equivalent linar approach: frequency domain (ProShake)

- Nonlinear approach: time domain

b,maxa

f ,maxa

P. Cacciola, E. Cascone, G. Biondi, Soil Amplification Factors for Soft Soil Deposits (Vs,30< 180 m/s) evaluated using a Data-Base of Italian Accelerograms,MERCEA'08

0 100 200 300 400V s (m /s)

P I= 1 5 % - Vs,30

= 1 7 9 m /sP I= 3 5 % - V

s,30= 1 5 1 m /s

P I= 6 5 % - Vs,30

= 1 0 3 m /s

zb= 3 0 m

zb= 6 0 m

zb= 9 0 m

zb= 1 2 0 m

G pK OCRp p

•The profiles of G0 were evaluated using the following relationship (Rampello, Silvestri, Viggiani 1994):

p’= mean effective pressure

OCR = overconsolidation ratio

pr = 1 kPa reference stress

P. Cacciola, E. Cascone, G. Biondi, Soil Amplification Factors for Soft Soil Deposits (Vs,30< 180 m/s) evaluated using a Data-Base of Italian Accelerograms,MERCEA'08

max ( )max ( )

••The study has been carried out through a modified version of ShaThe study has been carried out through a modified version of Shake 91ke 91

••The nonThe non--linear relationship between linear relationship between aaff,max,max and and aabb,max,max points out that the points out that the selection of constant values of the soil amplification factor inselection of constant values of the soil amplification factor in some cases some cases may be misleadingmay be misleading

0 0.1 0.2 0.3 0.4 0.5ag (g)

PI=65%zb=30mzb=60mzb=90mzb=120m

0 0.1 0.2 0.3 0.4 0.5

ag (g)0.0

0 0.1 0.2 0.3 0.4 0.5ag (g)

ab,max

ab,max ab,max

Cyclic behaviour of soils via the Preisach formalism

( ) ( ) ( )ZM Hτ γ = τ γ + τ γ

• For nonlinear time-domain wave propagation study pertinent cyclic model of soil have to be selected.

hysteretic counterpartZero memory nonlinear term

1 if or and decreasing( )

1 if or and increasingxα β

+ γ > α γ > β γτ = − γ < β γ < α γ

αβτ

γαβ

,( ) ( , ) ( )H d dα βα≥β

τ γ = µ α β τ γ α β∫∫

• hysterons (relay or hysteretic operator)

• According to the Preisach formalism (Preisach, 1935) hysteretic loops can be considered as a superposition of an infinite set of simplest hysteresis operatorsscaled by an appropriate weight function ( , )µ α β

Geometric interpretation of the Preisach formalism

( ) ( )

( ) ( , ) ( , )

2 ( , ) ( , )

d d d d

τ γ = µ α β α β − µ α β α β

= µ α β α β − µ α β α β

∫∫ ∫∫

( )S + γ

( )S − γ

( )L γ

Pα = α

Pβ = β

( )tγ

,j jα β

αβτ

the Preisach Plane

D• For a given time instant the domain can be subdivided in two sub-domains: the domain encompassing the set of the relay operators in the+1 status, and the domain encompassing the set

of the relay operators in the -1 status

P. D. Spanos, P. Cacciola, G. Muscolino, Stochastic averaging of Preisach hysteretic systems, Journal of Engineering Mechanics (ASCE), 130( 11), pp. 1257-1267, 2004.P. D. Spanos, P. Cacciola. and J. Redhorse, “Random vibration of SMA systems via Preisachformalism”, Nonlinear Dynamics 36, 405–419, 2004.P.D.Spanos, A.Kontsos and P.Cacciola, "Steady-State Dynamic Response of Preisach

Hysteretic Systems" Journal of Vibrations and Acoustics, 2006

Why Preisach ?

Extremely versatile: Used fromFerromagnetism to SMA

Capture minor loops

Robust

Enjoy of the Representation theorem

-1 0 1

-4 -2 0 2 4

-1 0 1

P. Cacciola, G. Biondi, E. Cascone, Wave propagation through Preisach formalism, TheTwelfth International Conference on Civil, Structural and Environmental Engineering Computing Funchal, Madeira, Portugal 1-4 September 2009

( ) ( )34

γ ξ γπχ = γ

• First application of Preisach formalism to cyclic modeling of soils

• Commonly cyclic behavior of soil is represented in terms of modulus reduction and dumping curve

• Through a procedure of harmonic balance the model parameters have been found in closed form

• Exact matching of the damping

1( ) ( sin )sine c cc

γ = τ γ ϑ ϑ ϑπγ ∫

1( ) ( sin ) cos2 ( )e c c

ξ γ = τ γ ϑ ϑ ϑγ π γ ∫

( )( ) 2 ( )cc

GG d dG α β

α≥β

γτ γ = γ + χ γ γ − χ τ γ α β

∫∫

P. Cacciola, G. Biondi, E. Cascone, Wave propagation through Preisach formalism, TheTwelfth International Conference on Civil, Structural and Environmental Engineering Computing Funchal, Madeira, Portugal 1-4 September 2009

0.0001 0.001 0.01 0.1 1 100

-1 0 1

-40000

-20000

-0.1 0 0.1

-0.01 0 0.01

-0.001 0 0.001

γ γ γ γ

τ τ τ τ

PI 100 %=50

-1 0 1

1cγ =

0.1cγ =

0.5cγ =

The proposed hysteretic model has been calibrated to match modulus reduction and damping curves proposed by Vucetic and Dobry (1981)

( )( )bb

τ γ = γ

ξ = =π π γ

•After defining a suitable cyclic model wave propagation can be carried out in time domain

•Coherence function (???)

Random response

Once defined reliably the stochastic seismic actions the structural response can be determined through:

Stochastic Calculus (linear behaving systems: Lyapunov equation; weak nonlinear systems: statistical linearization; nonlinear systems: Itocalculus, Fokker-Plank equation, ad hoc solutions, few dofs)

Monte Carlo Simulation (to date the unique universalmethod for coping with strong nonlinear problems)

Random response via modal analysis

- Seismic analysis of MDoF linear systems is usually performed adopting the well-known modal analysis along with modal truncation of higher modes.

- In deterministic case the accuracy of the modal analysis can be tested by evaluating the so-called “modal mass”.

- In stochastic analysis, to date, it is not possible to quantify the error due to modal truncation. Modal participation factors cannot be used directly for the stochastic analysis (Cacciola et al. Probabilistic Engineering Mechanics, 2007). For large FE modeled structural system the modal correction methods have to be introduced to improve the accuracy of the response.

- For both deterministic and stochastic input the simplest and most efficient correction method is the so-called “Mode Acceleration Method” (MAM).

- The MAM evaluates the contribution to the response of neglected higher modes in pseudo-static form (Maddox 1975).

Mode Acceleration Method (MAM)

By applying modal correction methods the response can be evaluated as the sum of

two contributions:

Mode Acceleration Method (MAM) – Maddox (1975). It is the simplest one. The corrective term is obtained considering in pseudo-static form the contribution of neglected higher modes:

( ) ( ) ( )PSˆt t t= + u uu

( ) ( )ˆ ˆˆ t t=u qΦ ( )PS tu so - called corrective term

( ) ( ) ( )-1 2 TPS g g

ˆˆ ˆt u t u t− = − − = u K M b && &&Φ Ω Φ τ

By applying the MAM in stochastic analysis it needs to evaluate the covariance matrix:

( ) ( ) ( ) ( ) ( ) ( )2ˆ E (ˆ ˆ ) E ( ˆ) E E ( )E TT T Tgg

Tgut t t t t tu t ut t= ++ + bb bu bu u u u u&& && &&

Covariance matrix of the response

ˆˆˆ ˆ ˆ ˆ( ) ( ) ( ) ( );

( ) 2 ( ) ( ) ( ).g

k k k k k k g

t t t u t

q t q t q t p u t

+ ξ ω + ω =

&& & &&

Ξq q q pΩ2( ) ( ) ( ) ( )gt t t u t+ + = −Mu Cu Ku M && & &&τ

( ) ( )ˆ ˆˆ t t=u qΦ

10( , ) ( ) ( ) ( , ) ( ) ( )( , ) ( ) ( )

u j jj

NRu j j

jG t a t GG t aG t Ga t G tω ωω ωω ω

= + = + =∑ ∑&&&&& &

( ) ( ) ( ) ( )

( ) ( ) ( )

*, , 0,

, 1 1 1 10

E , , ( )d ( )

, exp ( ) d .

k k k j j j k kj

k j k j

q t q t p p H t H t G p p t

H t h t i a

ω ω ω ω λ

ω τ ωτ τ τ

= − −

∑∫

l l l l l

ˆ ˆˆ ˆˆ ˆE ( ) ( ) E ( ) ( ) E ( ) ( )m m

T T T Tk k

t t t t q t q t= =

= = ∑∑u u q q l ll

Φ Φ φ φ

CLOSED FORM !

COVARIANCE MATRIX OF THE RESPONSE

Cross-covariance terms in the fully non-stationary

modelling

Covariance matrix of the response with MAM

( ) ( ) ( ) ( ) ( ) ( ) ( )-1 2 TPS g g

ˆˆ ˆˆ ˆ ˆt u tt t t t u t− + − −= = = u u u K Mu b u&& &&Φ Ω Φ τ +MAM:

MODIFIED COVARIANCE MATRIX OF THE RESPONSE

( ) ( )

( ) ( ) ( )

( ) , e

E ( )) ( ) ( )d

( ) xp d .

, exp d

E ( ) ( )

E ( ) (( ) ) .

q t p H t i t

u t a t G

= ω ω ω

= ω − ω

∫∑

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

ˆ ˆ ˆ ˆE E

ˆ ˆˆ ˆ E ( ) ( )

E ( ) E ( )

E ( )( E ( ))

T Tk k

u t uu t

t t t t t t

t t q u ttt qt= =

= ∑ ∑

u u u u + b bu u

q +bq +b b l ll

&& &&Φ

Φ φ φ

Input-output cross- covariances

Non-Geometric Spectral momentsIn stochastic analysis often the spectral moments have to be evaluated in order to perform the reliability analysis of structural systems.NON-GEOMETRIC SPECTRAL MOMENTS OF THE RESPONSE

( ) ( )

0, 0,1 1 1

1, 1,1 1 1

2, 2, 2,

2E( ) ( ) ( ) ;

( ) ( ) (

2 Re E ( )

Im E ) ;

ˆ ˆ ˆ( ) ( ) E

i i i i

u u k ik i k ik kk km m m

u u k ik i k ik

u u u u i i k ik

t p p t q t

t t u t u t p

φ φ φ

λ = λ +

λ = λ −

λ ≡ λ = = λ

∑∑ ∑

+l l ll

l l l l ll l

k = =∑∑

21, 1,

c,0, 0, 2,

( ) ( )( ) ; ( ) 1 .

( ) ( ) ( )i i i i

i i i i

i i i i i i

u u u uu u u u

u u u u u u

t tt t

t t tλ λ

ω = δ = −λ λ λ

Central circular frequency Bandwidth parameter

Numerical Application

The structure possesses 722 frame elements with 819 degrees of freedom. Mass is assumed lumped at each node. Furthermore, the structure is assumed classically damped with a modal damping ratio assumed constant for each mode.0 0.05ξ =

The structure undergoes to forced vibration due a ground motion acceleration acting in x-direction modeled as a Gaussian fully non-stationary spectrum compatible process.

Numerical ApplicationComparison of the stochastic response obtained by the proposed procedure with the results from MCS using 1000 samples. The modal correction method give a sensible contribution to improve the accuracy of the bottom nodes, whose response is affected by higher order modes. On the other hand the response of top nodes is accurately determined without any correction being governed by first two lower modes only.

0 10 20 300

4E-010

8E-010

1.2E-009

1.6E-009

0 10 20 300

2E-005

4E-005

6E-005

8E-005

2 modes

2 modes + correction

two modes

two modes + correction

16 modes + correction

16 modes

Fully non stationary input Quasi stationary input

P.Cacciola, N.Maugeri, G.Muscolino, A modal correction method for non-stationary random vibrations of linear systems, Probabilistic Engineering Mechanics, 22, 170-180, 2007

32.0 m

Moment resisting frame system.

1 1,87 76,22

2 5,61 89,21

3 9,77 93,27

jth frequency [rad sec-1]Modal

Partecipating Mass Ratio [%]

4 13,82 95,39

0 20 40 60

MDM (all modes)MDM (one mode)MDM (four modes)SMAM (one mode)

0 20 40 60

( )225u

σ ( )225u

[ ]t s [ ]t s

0 20 40 60

2E+013

4E+013

6E+013

8E+013

1E+014

0 20 40 60

4E+014

8E+014

1.2E+015

[ ]t s [ ]t s

0 10 20 30 40

0.1MDM (all modes)MDM (one mode)MDM (four modes)SMAM (one mode)

( )225u

[ ]t s0 10 20 30 40

0.03MDM (all modes)MDM (one mode)MDM (four modes)SMAM (one mode)

( )225u

[ ]t s

0 10 20 30 40

0.0E+000

2.0E+013

4.0E+013

6.0E+013MDM (all modes)MDM (one mode)MDM (four modes)SMAM (one mode)

0 10 20 30 40

0.0E+000

2.0E+014

4.0E+014

6.0E+014

8.0E+014

1.0E+015MDM (all modes)MDM (one mode)MDM (four modes)SMAM (one mode)

[ ]t s [ ]t s

5 10 15 20 25

MDM (one mode)MDM (four modes)SMAM (one mode)

( )u %25

[ ]t s5 10 15 20 25

( )[ ]

[ ]t s

( )%Vε

5 10 15 20 25

40MDM (one mode)MDM (four modes)SMAM (one mode)

[ ]t s5 10 15 20 25

5.0E+006

1.0E+007

1.5E+007

2.0E+007

( ) [ ]V t Nµ

[ ]t s

10 m10 m

0 5 10 15 20 250.0E+000

1.0E-005

2.0E-005

3.0E-005

4.0E-005

5.0E-005MCSMDM (two modes)MDM (five modes)SMAM (two modes)

( )2X,28u

[ ]t s

0 5 10 15 20 250

0.0005

0.0015 MCSMDM (two modes)MDM (five modes)SMAM (two modes)

( )2Y,28u

[ ]t s

0102030405060708090

MDM(five modes)

MDM(two modes)

SMAM(two modes)

(%)CPU∆

Stochastic Seismic Analysis: Stationary response (frequency domain)

Equation of Motion: linear behaving structure( ) ( ) ( ) ( )gt t t U t+ + = −MU CU KU M&& & &&τ

• Modal approach ( ) ( )t t=U QΦ

• Equation of Motion in Modal Subspace

( ) ( ) ( ) ( )2gt t t U t+ + =Q Q Q p&& & &&Ξ Ω

• “Mean” Peak Displacement /Response parameter

0, 1, 2, 0,( , 0.5) ( , 0.5; , , )r r r r rr s U s U U U UU T p T pη λ λ λ λ= = =

( )1 20 1 20 5 2ln 2 1 exp ln 2

r r r r r r r

.U s ,U ,U ,U U U U(T , p . ; , , ) N N

η = λ λ λ = − −δ π 1 1r j k

i,U r, j r, j i,Q Qj k

φ φ= =

λ = λ∑∑

• Peak factor •Spectral moments

12, 1,

0, 0, 2,

ln ; 12

U UsU U

TN p −λ λ= − δ = −

π λ λ λ*

( )( ) ( ) dj k gQ Q j k j k Up p H H G ωλ ω ω ω

= ∫ &&

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