Franch in Thesis

8/12/2019 Franch in Thesis

1/147


2/147

.


3/147

To Fidelia,

my sister Chiara,

my mother Gianna

and my father Roberto.


4/147

.


5/147

Contents

List of Figures v

List of Tables ix

List of Symbols xi

Abstract xiii

Acknowledgements xv

1 INTRODUCTION 1

1.1 Requirements of a reliability method for seismic risk assessment . . . 2

1.2 Engineering methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 State-of-the-art methods . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 FORM SOLUTION OF RANDOM VIBRATION PROBLEMS 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Models of the seismic action . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Random process modelling . . . . . . . . . . . . . . . . . . . . 13

2.2.1.1 Stationary models: the Kanai-Tajimi and Clough-

Penzien models . . . . . . . . . . . . . . . . . . . . . 13

i


6/147

2.2.1.2 Non-stationary models: periodic processes . . . . . . 16

2.2.1.3 Non-stationary models: Pulse-type processes . . . . . 19

2.3 Pulse-type discretization as a key to problem reformulation . . . . . . 22

2.3.1 The elementary excursion event for linear and non-linear systems 23

2.3.2 Design-point excitation . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 From the elementary excursion events to the first excursion

event: non-linear systems and mean out-crossing rate . . . . . 31

2.3.4 Linear systems: the simulation alternative . . . . . . . . . . . 36

3 LIMIT-STATE FUNCTION GRADIENT: RESPONSE SENSITIV-

ITY ANALYSIS 41

3.1 Limit-state functions defined in terms of a finite element model response 41

3.2 Equations of the response sensitivity . . . . . . . . . . . . . . . . . . 43

3.3 Shear-type structures with differential laws . . . . . . . . . . . . . . . 48

3.3.1 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 Differential model of hysteresis: Bouc-Wen . . . . . . . . . . . 49

3.3.3 Sample sensitivity time-histories . . . . . . . . . . . . . . . . . 56

3.4 Fiber models for reinforced concrete frames . . . . . . . . . . . . . . . 58

3.4.1 Displacement (Stiffness) Based Elements . . . . . . . . . . . . 60

3.4.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1.2 Response Sensitivity . . . . . . . . . . . . . . . . . . 61

3.4.2 Force (Flexibility) Based Elements . . . . . . . . . . . . . . . 63

3.4.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.2.2 Response Sensitivity . . . . . . . . . . . . . . . . . . 63

3.4.3 Constitutive Derivatives . . . . . . . . . . . . . . . . . . . . . 65

ii


7/147

3.5 Stress gradient for non-differential constitutive laws . . . . . . . . . . 69

3.5.1 Bilinear hysteretic law . . . . . . . . . . . . . . . . . . . . . . 71

3.5.1.1 Sample sensitivity time-histories . . . . . . . . . . . 73

4 APPLICATIONS 77

4.1 Simple model of a reinforced concrete building . . . . . . . . . . . . . 77

4.1.0.2 Ground motion . . . . . . . . . . . . . . . . . . . . . 78

4.1.0.3 Performance criterion . . . . . . . . . . . . . . . . . 79

4.1.0.4 Critical or least favorable excitation . . . . . . . . . 80

4.1.0.5 Effect of structure inelasticity and randomness . . . . 82

4.1.0.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 Reinforced concrete bridge . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.0.7 Bridge description and finite element modelling . . . 85

4.2.0.8 Ground motion and deterministic response . . . . . . 86

4.2.0.9 Results . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.0.10 Design point excitations . . . . . . . . . . . . . . . . 89

5 CONCLUSIONS 91

5.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Indications for future research . . . . . . . . . . . . . . . . . . . . . . 93

A RELIABILITY THEORY ESSENTIALS 95

A.1 Time-invariant reliability methods . . . . . . . . . . . . . . . . . . . . 95

A.1.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A.1.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

A.1.2.1 Montecarlo method . . . . . . . . . . . . . . . . . . . 97

iii


8/147

A.1.2.2 Importance sampling method . . . . . . . . . . . . . 98

A.1.3 FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.1.3.1 Component problem . . . . . . . . . . . . . . . . . . 100

A.1.3.2 System problem: general. . . . . . . . . . . . . . . . 103

A.1.3.3 System problem: FORM approximation. . . . . . . . 106

A.2 Random vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A.2.1 Random processes . . . . . . . . . . . . . . . . . . . . . . . . 107

A.2.1.1 Random process definition . . . . . . . . . . . . . . . 107A.2.1.2 Random process probabilistic characterization . . . . 107

A.2.1.3 Stationary (or homogeneous) random process . . . . 110

A.2.1.4 Spectral decomposition of a stationary random process111

A.2.1.5 Poisson counting process . . . . . . . . . . . . . . . . 112

A.2.2 Input-output relations in the frequency-domain . . . . . . . . 114

A.2.2.1 Deterministic input . . . . . . . . . . . . . . . . . . . 114

A.2.2.2 Random input . . . . . . . . . . . . . . . . . . . . . 116

A.2.2.3 Two related random processes . . . . . . . . . . . . . 118

A.2.3 Crossing theory . . . . . . . . . . . . . . . . . . . . . . . . . . 119

BIBLIOGRAPHY 121

iv


9/147

List of Figures

2-1 Clough-Penzien power spectra for different soil conditions. . . . . . . 15

2-2 Left: Discrete Clough-Penzien one-sided power spectrum for medium

soil conditions; Right: Autocorrelation function corresponding to the

discrete PSD beside. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2-3 Discrete Evolutionary one-sided power spectrum for medium soil con-

ditions (Clough-Penzien stationary spectrum modulated by an expo-

nential envelope). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2-4 Random process decomposition. . . . . . . . . . . . . . . . . . . . . . 24

2-5 Process excursions as half-spaces: (a) exact for linear systems; (b)

approximate for non linear systems. . . . . . . . . . . . . . . . . . . . 26

2-6 Design point for a linear system under stationary Gaussian white

noise excitation: (a) unit-impulse response function; (b) variance of

the response; (c) design-point excitation; (d) design-point response. . 30

2-7 White noise sample.

31

2-8 Reliability index evolution with time. . . . . . . . . . . . . . . . . . . 31

2-9 Crossing rate as parallel event (FORM approximation). . . . . . . . . . . 35

2-10 Magnitude crossing as up-crossing ofbk plus down-crossing ofbk. . . . 37

2-11 Sampling densities: (a) Proposed ISD; (b) Multimodal Gaussian. . . . 38

v


10/147

2-12 Convergence of probability estimate with number of trials (Results

for a six-story three-bay moment resisting frame under seismic load

reproduced from the example application in [4]), the circles indicate

Montecarlo results for 10,000 samples. . . . . . . . . . . . . . . . . . 40

3-1 Shear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3-2 Hysteresis loops for the Bouc-Wen model: for all cases the hardening

ratio is = 0.05, yield deformation dy = 0.2. (a) Reference case =

= 0.5 and variable m; (b) = 0.0 and variable (no hysteresis);

(c) = 0.0 and variable (no pinching); (d) variable A(strength). . 50

3-3 Hysteresis loops for the Bouc-Wen-Baber model: for all cases the

hardening ratio is = 0.05, initial yield deformation dy = 0.2. (a)

Stiffness and strength degradation due to A = 0.02; (b) Strength

degradation due to dy = 0.02; (c) Strength degradation due to =

0.02; (d) Unsymmetrical yielding due to = 0.4 . . . . . . . . . . . . 52

3-4 From left to right: base acceleration, displacement response, force-

deformation loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563-5 Displacement response sensitivity with respect to material parame-

ters (initial stiffness k, hardening ratio , yield deformation dy) for

integration time step: (a) t= 0.02sec. . . . . . . . . . . . . . . . . 57

3-6 Displacement response sensitivity with respect to material parame-

ters (initial stiffness k, hardening ratio , yield deformation dy) for

integration time step: (a) t= 0.005sec. . . . . . . . . . . . . . . . . 57

3-7 Displacement response sensitivity with respect to various ground ac-celeration pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3-8 (a) Complete system; (b) Basic system. . . . . . . . . . . . . . . . . . 59

3-9 Fiber discretization of a RC beam-column cross-section. . . . . . . . . 67

3-10 Hysteretic bilinear model parameters. . . . . . . . . . . . . . . . . . . 71

vi


11/147

3-11 Response of a SDOF oscillator to the first 10sec of El Centro 1941

acceleration record. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3-12 Sensitivity of the displacement response of the SDOF oscillator of

Figure(3-11) to material parameters E, Eh and fy. . . . . . . . . . . . 74

3-13 SDOF oscillator of Figure(3-11) subject to modulated white noise rep-

resented by a train of random pulses: sensitivity of the displacement

response to three of these pulses. . . . . . . . . . . . . . . . . . . . . 74

4-1 Shear model of a building. . . . . . . . . . . . . . . . . . . . . . . . . 78

4-2 Ground acceleration sample.

79

4-3 Hysteretic response under the input in Figure(4-2). . . . . . . . . . . . . 79

4-4 Reliability index versus time for a fixed value of PGA=0.2g and floor

ductility= 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4-5 Reliability index versus ductility for a fixed value of PGA=0.2g andtime t = 4sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4-6 Design point excitationfor a fixed value of PGA=0.2g, floor ductility= 3

and time t = 5sec.

81

4-7 Design point response at t = 5sec, PGA=0.2g, ductility capacity = 3:

thick line is 1st floor, thin lines the other floors. . . . . . . . . . . . . . . 81

4-8 Reliability index versus ductilityfor a fixed value of PGA=0.2gand time

t= 4sec: comparison between elastic and inelastic structural behavior for

deterministic(DT) and random(RT) threshold. . . . . . . . . . . . . . . 82

4-9 Reliability index versus timet for deterministic ductility capacity = 2. . 83

4-10 Reliability index versus ductility at time t = 4sec. . . . . . . . . . . . . 83

vii


12/147

4-11 Mean down-crossing rate versus time t for the inelastic structure and for

the 1st floor drift threshold 3dy.

84

4-12 Mean down-crossing rate versus time t for the tangent elastic structure

and for the 1st floor drift threshold 3dy. . . . . . . . . . . . . . . . . . . 84

4-13 Bridge layout: (a) general view; (b) deck cross-section; (c) pier cross

section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4-14 Bridge finite element model. . . . . . . . . . . . . . . . . . . . . . . . 87

4-15 Sample response of the bridge under base excitation. . . . . . . . . . 874-16 Sample response sensitivity of the bridge pier top displacements with

respect to some of the random acceleration pulses in the train of

Figure(4-15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4-17 Time-evolution of the reliability index . . . . . . . . . . . . . . . . . 89

4-18 Design point excitations for different threshold values at time t =

4sec, deterministic mechanical parameters. . . . . . . . . . . . . . . . 90

A-1 Random spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A-2 Basic system arrangements: (a) series and (b) parallel. (c) example

of general s y s tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A-3 FORM approximation (3 components): (a) series ; (b) parallel; (c)

general system, two cut-sets (g1,g2) and (g2,g3). . . . . . . . . . . . . 106

A-4 Filtered white noise PSD. . . . . . . . . . . . . . . . . . . . . . . . . 119

viii


13/147

List of Tables

3-1 Summary of displacement sensitivity computation for the Force and

Stiffness formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3-2 Derivatives of the stress increment with respect to n and . . . . . . 73

4-1 Floor yield force and drift. . . . . . . . . . . . . . . . . . . . . . . . . . 78

4-2 Upper bound to failure probability: comparison with Montecarlo simulation. 85

A-1 Examples of random processes/fields. (S) scalar, (V) vector, (D)

discrete, (C) continuous. . . . . . . . . . . . . . . . . . . . . . . . . . 108

ix


14/147


15/147

List of Symbols

The large number of quantities involved requires developing a rational notation to

be used consistently throughout. Since the study integrates tools coming from the

two different disciplines of reliability and mechanics, and some symbols are used

with different meaning in the two contexts, it has been chosen to keep the meaning

of the symbols as they are in one discipline and to change symbols from the other

when necessary. Specifically, the meaning of symbols as they are in mechanics is

kept unvaried and the notation used in reliability has been consequently adapted.

As usual, bold face indicates vector quantities and, among such quantities, vectors

are lower case and matrices capitals.

- ttime

- xi,x spatial coordinates

- i, natural coordinates

- U(t, x),U (t, x) displacement field

- u(t),u(t) nodal displacement

- u(t),u(t) nodal velocity u/t

- u(t),u(t) nodal acceleration 2u/t2

- q(t),q(t) modal displacement

- N(x) displacement interpolation function U(x, t) =N(x)u(t)

xi


16/147

- (t, x), (t, x) infinitesimal strain xU(x, t)

- a(x) =xN(x) strain-displacement function(x,t)=B(x)u(t)

- p(t), p(t) (external or applied) force

- pr(t), pr(t) internal or resisting force

- R(t), R(t),force residual or unbalanced forces

- (t),(t) stress

- h(t), h(t) internal history variables

- , random variable in the original space

- z,z correlated standard normal random variables

- y,y uncorrelated standard normal random variables

- v(t),v(t) (nodal) displacement sensitivity u/h

- v(t),v(t) (nodal) velocity sensitivity u/h

- v(t),v(t) (nodal) acceleration sensitivityu/h

- h(t)h

sensitivity history variables

- g() limit-state function in the original/physical space

-g() limit-state function gradient in the original/physical space

- G(y) limit-state function in the standard normal space

-G(y) limit-state function gradient in the standard normal space

- Jyjacobian of the transformation between original and standard normal space

xii


17/147

Abstract

In the relatively wide spectrum of reliability methods for seismic safety assessmenttwo distinct classes can be distinguished: that of engineering methods and that

of rigorous ones. Methods belonging to the former represent an approach to the

method formulation from the professional engineers side. In the trade-off between

the opposite requisites of practical usability and theoretical rigour, they do privilege

the first. The results they yield, though subject to certain limitations, are usable

in practice and arrived at with an acceptable computational effort. Methods of

the second class represent the complementary approach, looking for solutions that

reduce the onerosity of in general highly resource-consuming procedures withoutsacrificing the rigour coming from a sound reliability foundation. In most cases

these methods, when applied to real-scale structures, still imply a prohibitive cost

in terms of computational resources and knowledge to be acquired.

Persuaded that only a conceptually rigorous approach can provide a general

solution to the problem, in this thesis a recently proposed rigorous method is

extended towards practical applicability. Characterising trait of the method is that

of reducing a random vibration problem to the form that typically arises in the con-

text oftime-invariantstructural reliability. The mean crossing-rate of the dynamicresponse out of a safe domain is determined using FORM. A central role is played

by the limit-state function and its gradient. The computation of the latter accounts

for most of the resource consumption of the procedure. Optimization of the gradient

evaluation requires the development of specific dynamic response sensitivity analysis

tools for each mechanical model. In the context of the direct differentiation method

(DDM) beside the equations of motion one has to integrate an adjoint set of equa-

tions in the response sensitivity. Specific stress and resisting forces conditional

derivatives have to be developed to form the right-hand side of the adjoint system.

Previous applications of the method refer to SDOF dynamic systems or, for non-

seismic problems, to finite element models and specifically to plane-stress plasticity

problems. In the thesis the method applicability is extended to models of real

structures under seismic loading, and in particular to frame structures modelled

xiii


18/147

with fibre finite elements. The original contribution is the development of specific

response sensitivity tools for fibre elements. The result is implemented into an

existing code for three-dimensional inelastic dynamic analysis, for the stiffness and

the flexibility formulation as well, and for inelastic constitutive relations. This code

is then interfaced with another one that implements structural reliability methods,

and in particular FORM, to obtain estimates of the response distribution, mean

crossing-rate and first excursion probability.

Two applications demonstrate the method and its extension. The first refer to a

shear-type model of a building while the second to a three-dimensional fibre-model

of a bridge structure, both in reinforced concrete.

Notwithstanding the step forward towards the use of more realistic mechanical

models, the results achieved can not be considered yet as a satisfactory answer to

all the requisites of a method at the same time practical and rigorous. In particular,

the results have to be judged in the larger reference framework that includes classi-

cal simulation methods on one side and engineering methods on the other. The

formers are the only ones that, at least theoretically, can solve the problem taking

into due account all its factors, but they are not feasible of practical application to

real structures. With respect to these the method represents a decisive improvementin terms of resources required. With respect to the engineering methods unfortu-

nately, the comparison is still unequivocally in favour of the latters. On the other

hand it should be kept in mind that the method allows to consider sources of uncer-

tainty other than that coming from the earthquake, as for instance the randomness

in the mechanical parameters that influence the dynamic response of the structure,

which are not easily integrated in the engineering methods.

The most important still unsolved problem, in the author opinion, is that of

multiple collapse modes and, specifically, of the assessment of their mutual depen-

dence. With a notable exception of a recently proposed method, which is however

limited to linear elastic structures, no effective solution, i.e. other than Montecarlo

simulation, to the problem has been proposed. Future research will focus in this

direction.

xiv


19/147

Acknowledgements

My deepest and sincerest thanks go to my advisor, Prof. Paolo E. Pinto for theextraordinary attention he has dedicated to my work, for having always supported

as well as criticized me, for the example he has represented and the passion he

has transmitted to me. Equally for their scientific and moral rigor, and for their

great humanity, I wish to express my appreciation and thanks to Prof. Armen Der

Kiureghian and Prof. Filip C. Filippou, my mentors during my stay in Berkeley.

With these three persons, for the measure in which they contributed to my growth,

I will always be in debt.

Special thanks also to Prof. Fabrizio Vestroni, for having been always available

and understanding, to Prof. Giorgio Monti and Dr. Alessio Lupoi, for their friend-

ship and advices, and to Prof. Ove Ditlevsen, for having shown me how far I still

have to go.

My thanks and appreciation go to all those people that have made these three

years the richest and most intense of my whole life. They will be always in my

hearth.

xv


20/147

.

xvi


21/147

Chapter 1

INTRODUCTION

The uncertainty that characterises problems in earthquake engineering stems from

multiple concurrent sources and makes this discipline in this respect probably un-

rivalled among those of engineering. The intrinsic randomness in the occurrence

time and location of earthquakes, the vast uncertainty in predicting intensities and

characteristics of resulting ground motions, and the inability to accurately assess ca-

pacities of structures under cyclic loading, all compel us to make use of probabilistic

methods in order to consistently take into account the underlying uncertainties when

called to make quantitative assessments of structural safety.

This is the reason why, for both design and assessment, reliability-based ap-

proaches, as opposed to deterministic ones, are recognized as a superior mean for

measuring the performances of structures. This recognition, however, is not matched

by an equally wide diffusion in professional practice. For their potentialities to be

fully realised and hence their relevance in engineering practice to be enhanced, these

methods should meet a number of requirements that the present state-of-the-art still

does not allow to achieve, in the general case, in a way that is at the same timerigorous and practical.

1


22/147

2 1. INTRODUCTION

1.1 Requirements of a reliability method for seis-

mic risk assessment

The requirements that the ideal method should meet are implicitly defined through

the goal that the method should achieve, that of providing a realistic measure of the

total risk. Realism starts with the description of the seismic input, it involves then

the structural models and finally implies a degree of complexity and an amount of

computations which are acceptable to advanced engineering practice.

It is obvious that the confidence on the results of a seismic risk analysis largely

depends on a proper choice of the seismic action. It is equally clear that this choice

as well as that of the probabilistic model have to vary from case to case. Design

and assessment will have different requirements in this respect. Earthquake ground

motion can be described in one of the following ways, each suited for different

purposes: random processes, simulated accelerograms, recorded accelerograms and

synthetic accelerograms.

Random process modelling in conjunction with classical random vibration theoryhas been of invaluable help in the early stages of earthquake engineering in helping

understanding essential features of structural response in the elastic range. The

classicalmodels seem nonetheless unsuited for the purpose of realistic seismic risk

assessment of specific structures at specific sites, due to the averaged nature of

random processes models and the inability of random vibration theory in dealing

with the strong non-linearities exhibited by structural response in the vicinity of

collapse.

Simulated accelerograms are defined either as samples of a given random process,

i.e. compatible with a given power spectrum, or as ground motions that match an

assigned response spectrum. They can be used in non-linear dynamic analysis in

conjunction with realistic mechanical models and in this respect they remove the

limitation of random vibration theory. On the other hand, they still present the

drawback of being compatible with power or response spectra that come from the


23/147

1. INTRODUCTION 3

averaging of different spectra over wide areas, and are therefore unsuited for being

used for a specific site. They have their use in checking the internal consistency

of seismic design codes, i.e. whether structures designed with a ductility-reduced

elastic response spectrum and simplified analysis tools do actually satisfy, under the

test of a non-linear dynamic analysis , the performance objectives of the code.

State-of-the-practice of seismic risk assessment of important structures involves,

since a number of years now, the use of recorded accelerograms. In the last decade,

due to a number of large events that have struck developed regions of the world,

the number of collected records has increased enormously. It goes without saying

that, the main limitation of availability removed, recorded accelerograms representthe higher degree of realism achievable in terms of the action and, as for simulated

accelerograms, they can be used in conjunction with realistic mechanical models.

The subject of how to select and scale records for use at a specific site is widely

discussed in the literature [12][13].

Syntethic accelerograms, obtained from seismological modelling of the rupture

event at the source and of the stress waves propagation in the crust, are the new

frontier of earthquake models to be used in seismic risk [50]. Though much progress

has been achieved on these models, the number of applications in engineering arestill very few. There are nonetheless regions for which seismologists do consider

the seismotectonic features known in sufficient detail as to allow physically-based

accelerograms to be confidently generated and used.

Once a proper model of the action is chosen, the realism requirement involves a

corresponding choice for the model of the structure. One of the reasons why relia-

bility methods are being slow in gaining wider diffusion is in the contrast between

the mathematical complexity of the procedures, and the drastic simplifications thatquite often one has to introduce in the modelling of the structure. It is easily under-

stood how it can be difficult for an engineer confronted with a physically complex

problem to accept the idea that a probabilistic approach applied to a gross ideali-

sation of the problem can provide better information than a reasoned examination

of the results from a deterministic but accurate mechanical model.


24/147

4 1. INTRODUCTION

The ideal seismic risk analysis method should work jointly with advanced struc-

tural analysis methods, and with models of realistic dimensions. For the seismic

case the performance levels of interest for safety are those which are close to the

actual collapse of the structure. The structural model should be capable of cap-

turing all potential modes of failure, either displacement-controlled (ductile) ones

such as weak storeys failures and exhaustion of columns ductility, or force-controlled

(brittle) ones such as shear strength failures and joints failures.

Finally, on the reliability part, the method should be sophisticated enough as

to consistently treat the different sources of uncertainty (inherent randomness that

arises from natural variabilities in material as well as in load parameters, epistemic

uncertainty coming from incomplete statistical information, and model uncertainty

coming from the use of imperfect mathematical models to describe complex physical

phenomena), and to consider the possible dependence between all the failure modes.

The reader that wishes to have a comprehensive view of available reliability

methods in civil and in particular earthquake engineering can refer to three review

papers, [52][24][57], appeared in the last decade on the topic, the first two addressing

specifically seismic problems. In the following only a few methods, relevant to the

discussion and to this thesis, are considered.

1.2 Engineering methods

The ultimate purpose of a seismic risk analysis is to arrive at the (usually annual)probabilityPfof exceeding a given limit-state, or a set of limit-states, accompanied,

when feasible, by fractile values corresponding to selected confidence intervals on

the estimated value ofPf. This probability is called the risk.

In many past, as well current, applications, it has been customary to arrive at

the risk by first conditioning it on one parameter expressing the intensity of the


25/147

1. INTRODUCTION 5

seismic action. The final result is obtained then as

Pf= 0

Pf(I=i)dH(I=i) (1.2-1)

in which I is the selected intensity parameter, Pf(I = i) is the failure probability

as a function of I, also called fragility function, and H(i) = Pr{I i, 1year} isthe so-called hazard function characterising the seismic activity at the site under

consideration in terms of intensity. The distinction between fragility and hazard

dates back to the early phases of seismic reliability and was intended as a mean to

separate the tasks of the structural engineer and of the seismologist. It is important

to recognize, however, that the fragility function does not depend on the structuralproperties only but on all the characteristics of the seismic input except for the the

single scaling factor of the intensity. In a different environment, therefore, the same

structure would have a different fragility.

Most of the methods available are actually intended at evaluating the fragility

or, even more simply, the failure probability conditioned on one specific sample

accelerogram. Since the action variability is dominant in seismic problems this

result is clearly insufficient and has to be regarded as a step in a wider procedure in

which this latter variability has to be introduced subsequently.

Finally, a closer examination of the seismic risk assessment problem, as expressed

by Eq.1.2-1, allows a more clear appreciation to be gained on the importance of an

accurate estimation of the fragility term in the integral. In the most simple terms

Eq.1.2-1 is another version of the classical reliability problem formulation

Pf=

0

FR()fS()d (1.2-2)

where now the resistance distribution FR() = Pr{R } represents Pf() andfs() is the density of the loading and it is verifiable that when the variability of

one of the two terms (the load in this case) dominates, neither the variability of the

other term nor the precise form of its distribution have a significant influence onPf.

This consideration has originated in the last decade much research to investigate

under what conditions and with how much accuracy (or loss of it) the fragility could


26/147

6 1. INTRODUCTION

be evaluated using a drastically reduced number of samples, of the order of a few

tens. The main outcome of this research are two methods, [22] and [35][23], that

represent what could be called the advanced state-of-the-practice in seismic reliabil-

ity assessment. Their final result is different since one yields the total risk [22], while

the other gives the fragility function, but they have the common characteristic of

employing only basic probability theory notions and of working in conjunction with

state-of-the-art mechanical models for inelastic dynamic analysis of real-scale struc-

tures. These two traits, together with the extremely sparing number of structural

analyses required, make them very attractive for use in professional practice. In

particular the first one has been also cast in a partial safety factors format suitable

for use in design.

When compared with that of classical simulation methods, Montecarlo and its

variations, the computational demand of these methods is dramatically lower. The

main drawbacks are the inability to take into account the randomness of the struc-

ture and the dependence among different failure mechanisms that concur in causing

the structural collapse. Removal of these limitations is made difficult, if possible at

all, by the very nature of the methods and their underlying simplified probabilistic

setting. They have to be considered as a successful attempt to approach the idealmethod from the engineering side of the problem, meeting the requirement of accept-

able complexity and computational demand, and that of using realistic structural

models.

1.3 State-of-the-art methods

Some of the methods described in this section have the common feature of employing

a discretised form of the input to reduce a time-variant problem to a time-invariant

one, formulated in a space of random variables , and that can be solved by the well-

established methods of structural reliability [42][29][25][4][5]. Others use recorded

accelerograms and aim at finding Pf conditional on the record, [67], or treat the

record inside a response surface approach [62].


27/147

1. INTRODUCTION 7

Methods based on the FORM/SORM techniques have been proposed in the last

ten years by Der Kiureghian and others. One of these proposals presented in [67]

applies to a generally non-linear structure with random parameters, subjected to

a deterministic load history possibly modulated by some random parameter, for

example the intensity. Solution is sought to the problem commonly known as first-

excursion, i.e. the probability that a selected scalar response quantity exceeds a

given threshold in time interval D. To this purpose, a time independent limit-state

function is defined as the minimum value, over D, of the safety margin function

z(t, ) =r(t, ) s(t, )

where r ad s represent resistance and response, respectively, and they are both in

principle functions of time t and of the vector that in this case collects random

variables describing randomness of the structure. Discretisation in time is achieved

by considering the function z(t, ) at N time instants in D, so that the limit-state

function g() becomes g() = min{zn()}, where n indicates the generic instantwithin theNconsidered. Given a realization of the function z(t, ) can be calcu-

lated via structural analysis at all instants, and the minimum found at time instant

n = n, so that g() = zn(). In the search for the design point ofg one then

moves on the surface zn() = 0, but during the search it may well happen that

for an improved point i the instant of time when zn() is minimum changes and

hence the g-function changes in consequence. To account for this specific feature of

the problem a modified version of the usual search algorithm employed in FORM

analysis has been developed, suitably modified so as to be able to identify jumps

from one surface to another one and continue moving on the latter and possibly

on other ones until the design point is found. It is observed however that in this

problem more than one design point is significant since a number of almost equally

likely excursions at other time instants (as it is usually the case with earthquake

induced oscillations) may contribute to Pf. Failure probability is determined then

as a series system between the determined excursions but there is no guarantee that

all the significant design points have been found, thus the method yields only a lower

bound to Pf. This, together with the fact that the analysis has to be repeated for

many records, is the reason why research in this direction does not seem to have


28/147

8 1. INTRODUCTION

progressed further even if the method has been applied to realistic structures.

A second proposal by Der Kiureghian and others [42][29][28][25][40], as well asother two methods based on simulation proposed by Beck and others [4][5], em-

ploy fully non-stationary random process models (in particular realisations of these

processes in conjunction with non-linear dynamic analysis) and are based on the dis-

cretisation of the input process in terms of a train of random pulses. These pulses

are collected in the vector of the random variables together with the other vari-

ables that represent randomness in the structure (with the exception of the method

presented in [4]). The difference in the methods lies in the reliability method chosen

after the process discretisation to solve the resulting time-invariant problem.

Recognizing that simulation is the only reliability method that offers a rigor-

ous solution of the very complex first excursion problem, but also that classical

Montecarlo simulation requires a prohibitive number of analyses for estimating low

probabilities, even if in earthquake engineering problems these are much higher than

for non accidental types of action, Beck and others have proposed two enhanced sim-

ulation techniques. They both yield as final result a fragility curve. The fist one,

presented in [5], is based on the concept of subset simulation. The idea is that the

determination of the very low failure probability associated with an extreme eventcan be split into the computation of the failure probabilities of a sequence of failure

events characterized by much higher probabilities. IfFrepresents the failure region1of the event under consideration, letF1 F2 . . . Fm =Fbe a decreasing se-quence of failure events so thatFk =

ki=1 Fi, k = 1, . . . , m. For example, if failure

is defined as the exceedence of an uncertain demand D over a given capacityC, that

is Pr{D > C}, then a sequence of decreasing failure events can simply be defined asFi = {D > Ci}, where C1 < C2 < . . . < C. By definition of conditional probabilitythen one has

Pf=P(F) =Pm

i=1Fi

= PFm|m1

i=1Fi

Pm1

i=1Fi

=

=P(F1)m1

i=1P(Fi+1|Fi)

1The subspace of the random variable space that contains all the realizations of that corre-spond to non-satisfaction of the performance criteria.


29/147

1. INTRODUCTION 9

The idea is to estimatePfby estimating the conditionalPf,i. The advantage is that,

though Pf is small, by choosing the intermediate failure events appropriately, the

conditional probabilities can be made sufficiently large so that they can be evaluated

efficiently by simulation. For example, suppose that P(F1), P(Fi+1|Fi) 0.1, i =1, . . . , 5, thenPf 106 which is too small for efficient Montecarlo simulation. Theproblem of simulating samples according to the conditional distributions is solved by

the Metropolis method (Markov chain simulation). This method employs state-of-

the-art mechanical models, can take into account mechanical randomness and also

greatly reduces the number of simulations required. Nonetheless, even if well below

the millions of simulations required by plain Montecarlo to estimate probabilities of

the order of 106, the number of simulations is still in the order of a few thousands,

too far from the numbers that could be considered in engineering practice, especially

with real-scale structures.

The other simulation method, presented in [4], is a very efficient form of impor-

tance sampling. This method is discussed later on for its relevance to the subject

of the thesis. Here it is observed only that its advantages are the capability of con-

sidering as many failure mechanisms as needed and to properly take into account

their dependence, and the very low number of simulations required, in the orderof a few tens, as in the case of the state-of-the-practice methods mentioned in the

previous section. The usefulness of the method, however, is severely limited by the

hypothesis of linear structural behaviour, on which its superior efficiency is based,

and by the fact that it can not account for randomness in the structure.

The method presented in [42][29][28][25][40] uses again FORM/SORM to com-

pute failure probabilities. Here the usual algorithms for the search of the design

point employed for static problems can be used since there is no bouncing between

surfaces corresponding to excursion at different time instants, as it was the case forthe method in [67]. The method offers an upper bound to the first excursion prob-

ability using the crossing-rate theory. The final result of the method is a fragility

curve. Realistic structural models can be used while retaining computational effi-

ciency, provided the mechanical models are properly enhanced with capabilities

necessary to compute the gradients of the limit-state function. In particular, the


30/147


31/147

Chapter 2

FORM SOLUTION OF

RANDOM VIBRATIONPROBLEMS

2.1 Introduction

In this chapter a rather original, and recent, perspective on random vibration prob-

lems is presented. Problems formulated in terms of random processes in time are

transformed into discrete ones in terms of time-independent random variables. In

the vector space of these random variables geometric interpretations become possible

for events of interest defined in terms of the input/output processes. In particular,

random vibration of linear systems give rise to simple, linear, geometric shapes, such

as hyper-planes and wedges formed by their intersection.

The reformulation is based on a discretisation of the input process, a discrete

form for the output ones following immediately. Specifically, the ground motionrandom process, the input, is represented as a train of random pulses. This discrete

representation is introduced in Section 2.2 in the wider framework of random process

models of the seismic action.

The originality of the approach does not consist of seeking a discrete solution

to a continuous problem but in the fact that the considered discretisation reduces

11


32/147

12 2. FORM SOLUTION OF RANDOM VIBRATION PROBLEMS

the problem to a well-known form that arises in the context of structural reliability.

All the well-established methods of structural reliability can then be used to find

a solution, either exact1 for linear problems or approximate for non-linear ones. In

particular the methods based on the expansion of the limit-state surface around

the design point (FORM/SORM) are of interest in the present work, with specific

interest into non-linear problems.

Finding the design point becomes then a central part of the random vibration

problem. Algorithms that perform this search more effectively require the compu-

tation of the limit-state function gradient. When random vibrations of a complex

structure are of interest, the latter is commonly described by a finite element model.Computation of the gradient in this case, i.e. when the limit-state function is formu-

lated in terms of a numerically determined response, becomes a challenging problem

and accounts for most of the computational effort. This is the reason why the

problem is dealt with extensively in Chapter 3, which is entirely devoted to it.

2.2 Models of the seismic action

Regarding nonlinear dynamic analysis as the only acceptable mean to assess the

reliability of a reinforced concrete structure under seismic excitation, the latter must

be necessarily provided in the form of acceleration time-histories. These may belong

to one of the three broad categories below

- samples of a random process

- recorded (or natural) accelerograms

- accelerograms from seismological models (or synthetic)

The interest here, in view of the fact that the problem at hand is one of random

vibrations, is for those acceleration time-histories that belong to the first category.

The latter two classes constitute two very large worlds that will not be entered here

and are included only for completeness.

1Consistently with the discretisation.


33/147

2. FORM SOLUTION OF RANDOM VIBRATION PROBLEMS 13

2.2.1 Random process modelling

A basic need of the earthquake engineering community has always been that of defin-

ing realistic models of the seismic action for design purposes. Without seismological

models yet available to help for this task, engineers started to look at the records

that were rapidly accumulating, in search of characteristics of the ground motion

possessing a stable statistical nature (given of course some global earthquake data

such as the magnitudeM, distancedand recording site soil type S). The observed

statistical stability of the frequency content of the motions under similar conditions

ofM,d and Sis at the base of the idea of considering the accelerograms as samples

of random processes. Several stochastic models of varying degrees of sophistication

have been proposed.

The empirical approach based on the inspection of the available recorded ground

motion data base, has focused principally on the frequency content of the ground

motion, with due attention also paid to the modulation in time of such motion

and, to a much lesser extent, to the evolution with time of the frequency content.

This latter dependence of the frequency content on time, observed in real ground

acceleration records, originates from the complexity of the radiation of the seismic

waves from the source to the site, i.e. from the fact that P (primary/body), S

(secondary/shear) and Rayleigh-Love (surface) waves have different frequency con-

tent and arrival times at a given site since they propagate with different velocities

in the earth crust. Calibration of models that take into account this frequency-

content time-dependence versus statistical data has proven difficult and prone to

much uncertainty, so that statistically-based fully non-stationary stochastic earth-

quake ground motion models are not available. The earliest and still more widely

used models are stationary or, at most, amplitude-modulated random processes.

2.2.1.1 Stationary models: the Kanai-Tajimi and Clough-Penzien mod-els

The basic and, under the simplest of the assumptions, also sufficient description of

a random process is S(), its power spectral density function (PSD). A zero-mean


34/147


stationary Gaussian process is in fact completely defined once its PSD is specified.

Among idealized ground motion models based on empirical data the most-widely

used PSD form is that proposed by Kanai [38] and Tajimi [61] to take into account

the observed dependence of the power spectrum on local site soil properties and the

corresponding dominant frequency.

The proposed PSD coincides with that of the acceleration response process of a

SDOF oscillator with parameters g and g subjected to a white noise process of

intensityS0. This PSD is given by the productS0|H()|2 whereH() is the complexfrequency transfer function of the linear oscillator whose motion is governed by

ua+ 2ggur+2gur = 0

in whichuaand urare the oscillator absolute and relative displacement, respectively.

In the equation above one can either replaceuawithur+ug or urwithuaug gettingone of the two following equations

ua+ 2ggua+2gua = 2ggug+

2gug (2.2-1)

ur+ 2ggur+2gur = ug (2.2-2)

They lead obviously to the same frequency transfer function between the Fourier

transforms of ugand ua, once it is recalled that for the second the ground acceleration

has to be added

H() =2g + i2gg

2g 2 +i2gg (2.2-3)

Making use of this transfer function one gets for the Kanai-Tajimi density

SKT() =4g + 4

2g

2g

2

2

g 22

+ 42

g2

g2

S0 (2.2-4)

which with a suitable choice of the parameters g and g can be shown to match

with adequate accuracy empirical PSDs elaborated for different soil conditions. The

model has also a qualitative physical support, since it can be viewed as the PSD of

the acceleration process at the upper surface of a soil stratum, modelled as a linear

filter, having a broad-band excitation process at the bedrock.


35/147


0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

[rad/sec]

S()

stiffmediumsoft

Figure 2-1: Clough-Penzien power spectra for different soil conditions.

The use of the Kanai-Tajimi model, however, is limited to those cases when accel-

eration responses are of interest and the problem is not sensitive to the low-frequency

range of the excitation, since the corresponding ground displacement process PSD,given by Sd() = SKT() /

4, has infinite total power, with largest contributions

coming from the very low frequencies. To solve this problem Clough and Penzien [17]

have suggested the introduction of a second filter in series with the first one, with

parameters fandfappropriately chosen in order to filter out the low frequencies.

The corresponding PSD is

SCP() = |Hf()|2 SKT() = 4

2f 22 + 42f2f2

SKT() (2.2-5)

Frequently adopted values for the filters properties are: firm or stiff soil conditions

g = 5 rad/sec, g = 0.6, medium soil conditions g = 3 rad/sec, g = 0.4, soft

soil conditionsg = rad/sec,g = 0.2 and, for all soils, f= 1.6rad/sec,f= 0.6,

the resulting power spectra being shown in Figure 2-1 for a unit input white noise

intensity.


36/147


2.2.1.2 Non-stationary models: periodic processes

Periodic process are introduced as a possible mean to simulate those sample real-

izations of a random process that are needed in order to perform nonlinear dynamic

analyses. With reference to these processes the issue of inclusion of non-stationarity

in the ground motion model is also addressed.

Consider the process X(t) periodic of period T, defined as a linear combination

of harmonic functions with random coefficients in the form

X(t) =N

k=1

(Akcos kt+Bksin kt) (2.2-6)

where Ak,Bk are zero-mean, uncorrelated Gaussian variables with common variance

2k and k = k1 = k2/T. This process is Gaussian zero-mean and can be easily

shown to be stationary through its autocorrelation function RXX. Computing the

latter and taking its Fourier transform yields an expression for the process PSD as a

function of the variablesAk and Bk which can then be related to a known PSD, for

example to the Kanai-Tajimi or the Clough-Penzien one. The latter result allows

easy generation of samples according to a process with given PSD.

The autocorrelation function of the process X(t) is

RXX(t1, t2) = E [X(t1) X(t2)] =

= E

Nk=1

(Akcos kt1+Bksin kt1)Nh=1

(Ahcos ht2+Bhsin ht2)

=

=Nk=1

E

A2k

cos kt1cos kt2+ E

B2k

sin kt1sin kt2

=

=

Nk=1

2kcos k(t2 t1) =

Nk=1

2kcos k=RXX() (2.2-7)

which, as anticipated, shows that the process is stationary. The PSD of the process

is then given by

SXX() = 1

2

RXX() eid=

Nk=1

2k2

[(k ) +(k+)]


37/147


Due to the symmetry of the PSD it is easier to work with the so-called one-sided

PSD GXX() given by

GXX() =Nk=1

2k(k )

defined only on the positive portion of the frequency axis. The above relation shows

that the one-sided PSD is a discrete function made up of spikes of intensity 2k at the

frequencies k. Simulation of sample realizations reduces to the generations of 2N

zero-mean Gaussian random variables with variance given by 2k = GXX(k),

where GXX(k) is the one-sided form of the continuous target PSD, for example

that in Eq.(2-1), which is in this way approximated by its discrete counterpart.

0 10 20 300

2

4

6

[rad/sec]

G()

10 0 1020

0

20

40

60

[sec]

R()

Figure 2-2: Left: Discrete Clough-Penzien one-sided power spectrum for mediumsoil conditions; Right: Autocorrelation function corresponding to the discrete PSDbeside.

A number of variants of the processX(t) in Eq.(2.2-6) can be obtained by simpletransformations. These turn out to be more convenient for simulation purposes and

can be shown to have the same second-moment characterization. One of these

variants is

X(t) =Nk=1

Ckcos (kt+k) (2.2-8)


38/147


This form is easily shown to be equivalent to that in Eq.(2.2-6) by

Nk=1

Ckcos (kt+k) =Nk=1

[Cksin kcos kt+Ckcos ksin kt]

which implies that Ck =

A2k+B2k and k = arctan (Ak/Bk). IfAk, Bk are uncor-

related Gaussian, Ck and k are Rayleigh and Uniform, respectively. The autocor-

relation function of this process can be shown to coincide with that in Eq.(2.2-7).

Another process whose autocorrelation function is identical to the previous one is

defined as

X(t) =N

k=1

2kcos (kt+k) (2.2-9)

This last form has the great advantage of requiring only N random variables to

generate a sample of the process and this is the reason why it is the more widely

used for this purpose2.

One has to note that, even if the three processes in Eq.(2.2-6), (2.2-8) and (2.2-

9) have the same second-moment characterization, they are not probabilistically

equivalent since they have different distributions. While the process in Eq.(2.2-6)

is always Gaussian, the other two are only asymptotically Gaussian if N is large

(simulations show that N >100 makes the assumption of Gaussianity acceptable).

A straightforward way to introduce non-stationarity in the periodic processes

above is to modulate them with a deterministic envelope function as in

XNS(t) =e (t, ) X(t)

where the modulating function e (t, ) depends on both time and frequency as it

is the general case for fully non-stationary processes. Most often the information

available is not sufficient to establish a reliable dependence on the frequency and

e (t, ) is replaced by e (t). When the latter is a slowly varying function of time,the process above represents an amplitude-modulated harmonic function whose evo-

lutionary PSD can be shown to be (uniformly modulated process, [53], see Figure

2-3)

SNS(, t) = |e (t)|2 S()2Of course the degree of randomness is halved.


39/147


010

200 10 20 30

0

5

10

15

t [sec][rad/sec]

G(,t

)

Figure 2-3: Discrete Evolutionary one-sided power spectrum for medium soil condi-tions (Clough-Penzien stationary spectrum modulated by an exponential envelope).

Correspondingly, the process variance becomes

2X(t) =

SXX(, t) d=

0

GXX(, t) d =

0

e2 (t) GXX() d=20e

2 (t)

2.2.1.3 Non-stationary models: Pulse-type processes

Similarly to what has been done with periodic processes, a number of non-stationary

process models that retain the main idea of the Kanai-Tajimi one3 are now presented.

These are based on the mathematical concept ofrandom pulse train.

The physical basis underlying these models is in the mechanism of earthquake

generation: slips in a fault zone in the crust. When a slip is taking place and the

fault zone is being extended, the behaviour of the material in the vicinity of the

fault is obviously inelastic. However, away from the immediate vicinity, the energy

released and the associated particle motion is transmitted in the ground essentially

through linear stress waves. The typical presence of high-frequency components in

3The idea that the motion at the ground surface can be thought of as some broad band bedrockexcitation linearlyfiltered by the soil stratum above.


40/147


earthquake records also suggests that slips in a fault zone occur intermittently rather

than smoothly. A possible model is to assume that slips occur in independent short

spurts, almost impulsively [43]. This can be mathematically described quite simply

as

Xfault(t) =

N(t)k=1

Yk(t k) (x k) (2.2-10)

where (t k) and(x k) are Diracs delta in time and space which differ fromzero only at time k and location k (in the fault zone) of the occurrence of the

k-th pulse with random magnitude Yk. The magnitudes of the pulses are assumedto be independent but identically distributed. The number of pulses up to time t is

counted by the random process N(t). The process in Eq.(2.2-10), when the pulses

occurrence times are uncorrelated, is calledshot-noise. In what follows the stronger

assumption that the pulses occur independently, i.e. that N(t) is a Poisson process,

will be made.

A shot noise is non-stationary when either the average pulse occurrence rate

depends on time, or, when is constant, if the distribution of the Ys are time-

dependent. If neither of the two conditions above is verified, the process is stationary

and becomes a white noise process (as the bedrock broad-band excitation of the

Kanai-Tajimi model).

The ground surface motion related to the process above is obtained from the

Greens functiong (t k, x k) that gives the motion at time t and locationx (atthe ground surface) due to a unit-impulse at time k and location k (in the fault

zone)

X(t, x) =

N(t)k=1

Ykg (t k, x k) (2.2-11)

If the inverse Fourier transform of the frequency transfer function of the Kanai-

Tajimi model (see Eq.(2.2-3)) is used as Greens function in Eq.(2.2-11) (dropping

the dependence on the locations since only one input and one output locations,


41/147


bedrock and surface, are of interest) one gets

X(t) =

N(t)k=1

Ykha(t k) (2.2-12)

where the acceleration unit-impulse response function has the form

ha(t k) = 2gghd(t k) +2ghd(t k) =

=gegg(tk)

1 22g

1 2gsin gd(t k) + 2gcos gd(t k)

in which hd is the well-known displacement unit-impulse response function.

If the condition for the process to be stationary are satisfied the PSD of the

process in Eq.(2.2-12) approaches that of the Kanai-Tajimi model.

In particular in what follows a simplification of the described process is of interest,

i.e., a process for which the pulse arrival times are equally spaced

X(t) =Nk=1

Ykha(t k), k = (k 1) t (2.2-13)

and the pulsesYkare defined as integrals of a zero-mean Gaussian white noise process

W(t)

Yk =

k+1k

W(t) dt

It is immediately seen that the pulses are zero-mean uncorrelated Gaussian variables

with common variance given by

2Yk = E

k+1k

W(t1) dt1

k+1k

W(t2) dt2

=

k+1k

k+1k

E [W(t1) W(t2)] dt2dt1=

= k+1

k k+1

k

RWW(t2

t1) dt2dt1=

k+1

k k+1

k

2S0(t2

t1) dt2dt1 =

=

k+1k

2S0dt= 2S0t

Non stationarity can be reintroduced in this simplified model4 as it has been

done for periodic processes by means of an amplitude-modulating envelope function

4It is recalled that the starting model in Eq.(2.2-11) is fully non-stationary.


42/147


e(t). Making use of multiple envelopes and filters one can also model spectral non

stationarity with the process

X(t) =

Nfi=1

ei(t)

Nk=1

Ykha,i(t k)

(2.2-14)

It is finally observed that simulation of random process realizations according to

these pulse-train models reduces, as it was the case for periodic processes, to the

generation of a random vector of length N. While in both periodic and pulse-train

modelsNdetermines the richness of the process in terms of frequency content, N is

not the same for the two cases since for the pulse-train one it is also strictly relatedto the process sample duration.

The model in Eq.(2.2-14) represents the starting point for the reformulation of

the random vibration problem anticipated at the beginning of this chapter.

2.3 Pulse-type discretization as a key to problemreformulation

As anticipated at the beginning of this chapter, a new perspective on random vibra-

tions problems is gained once a discrete representation of the input random process

is available. In particular when this discrete representation is in terms of a vector of

standard normal random variables, many statistical quantities of interest in classical

random vibrations, such as the distribution of the process and its envelope, correla-

tion functions and statistics of crossing rates, can be given geometric interpretations

in the standard normal space.

When the structure is linear and the excitation or input process is Gaussian

the statistical quantities above have simple geometric forms as counterparts, and

the results found are consistent with the classical solutions. When the response

processes are non Gaussian either due to non linear structural behaviour or non-

Gaussian excitation, the geometric forms are not as simple and exact solutions are

not available. In this case, however, the well established tools of time-invariant


43/147


reliability can be used to achieve approximate solutions.

Possible approaches to the solution of random vibration problems are based onFORM/SORM techniques and on importance sampling. Before a closer analysis of

these is undertaken, geometric interpretations are given for a number of well-known

quantities of interest.

2.3.1 The elementary excursion event for linear and non-linear systems

Consider the ground acceleration random process as discretized in Eq.(2.2-13). Thisis only one of the possible alternatives for the discrete representation of a Gaussian

process. In general all such existing discrete representations can be expressed in the

form

X(t) =X(t) +Ni=1

si(t) yi= X(t) + sT (t) y (2.3-1)

where X(t) is the time-varying deterministic mean function of the process, thedeterministic functionssi(t) give the time-evolution of the process and the standard

normal random variablesyiits randomness. With reference to the process in Eq.(2.2-

13) one sees that the mean function is identically zero and si(t) =

2S0t ha(t ti) with ha(t) the acceleration unit-impulse response function of the linear filter

and

2S0t the standard deviation of the random pulses whose standardized

intensities are given by the random variables yi.

It is noted thatN is a measure of the resolution of the representation and that its

value, as well as the vector of deterministic basis functions s(t), depends on the cor-

relation structure of the process. Alternative representations differ in the choice of

the basis functions, for example for the Karhunen-Loeve expansionsi(t) =

ii(t),

where i, i(t) are thei-th eigenvalue and eigenfunction of the autocovariance func-

tion of the process[60], for an expansion in terms of trigonometric functionsone has

s(t) =T(t)L, where(t) is a vector of simple trigonometric functions while L is a


44/147


45/147


t, for the system starting at rest, is given by the Duhamels convolution integral

u (t) = t

0

X() h (t ) d= t

0

Ni=1

yisi()h (t ) d= Ni=1

yiai(t) =aT (t) y

(2.3-2)

where, having exchanged the order of summation and integration, the response

processi-th basis function has the form

ai(t) =

t0

si() h (t ) d

The interpretations given for the variance and the autocorrelation function of the

input process hold for the response process too. Other simple interpretations could

be easily given, for example for the input-output cross-covariance function.

Consider now the event of the process u(t) exceeding a predefined threshold bk

at time t = tk, written as{u (tk) bk}. From Eq.(2.3-2) it immediately followsthat all the possible input processes corresponding to this event are given by those

realizations ofy that satisfy the condition

bk

aT (tk) y

0 (2.3-3)

which is an half-space bounded by the hyperplane bk aT (tk) y = 0 of normalk =a (tk)/a (tk) and distance from the origink =bk/a (tk) =bk/u(tk). Thecondition in Eq.(2.3-3) represents the failure domain of the linear limit-state function

G(y) =bkaT (tk) yin the standard normal space and the exact probability contentassociated with this failure domain is well known to bePf= (k). This confirmsthe result already available from classical random vibrations, that the probability of

the zero-mean Gaussian process u(t) having variance 2u(t) of exceeding bk at time

tk is 1 ( bk0u(tk)).

The latter result is quite important. The event {u (tk) bk} is completely definedby the design point of the limit-state surface G(y) = 0. It is shown in the next

section how this design point is analytically known for linear systems under Gaussian

excitation. The geometry of this exceedance event at various time instants for the

respective, generally time-varying, thresholds is shown in Figure(2-5(a)).


46/147


Figure 2-5: Process excursions as half-spaces: (a) exact for linear systems; (b)approximate for non linear systems.

When theresponse is non Gaussian, for example due to a non Gaussian input or

to a non linearity of the system, the response representation in the form of Eq.(2.3-2)

has to be replaced by a more general non linear one such as

u (t) =u (y, t)

which has a geometric representation in the standard normal space other than an

hyperplane. In this case FORM or SORM techniques can be used to obtain an

approximate value of the probability associated with the event{u (tk) bk}, asshown in Figure(2-5(b)), but numerical solution of the constrained optimization

problem with constraint Gk(y) =bk u (y, tk) = 0 must be resorted to in order tofind the design point.

The usefulness of this detailed discussion on this excursion event, in the following

referred to aselementary, emphasizing the fact that it is an excursion at a fixed time

instant, as opposed to the more complex first5 excursionevent, will become clear in

sections 2.3.3 and 2.3.4.

5over a time interval of duration D


47/147


Of great interest is the so-called design-point excitationobtained as

X (t) =sT (t) y (2.3-4)

This input is, among all inputs that cause a response u(tk) greater than or equal

to the threshold bk, the most likely one. As it will be made clear in the following

paragraph, X(t) is a nearly harmonic, gradually intensifying function of time with

a frequency close to the fundamental frequency of the system. The nature of this

somewhat unnatural excitation has been extensively investigated, see for example

[31] and [40]. In the next section the design point excitation is discussed with

reference to linear systems.

2.3.2 Design-point excitation

As it has been anticipated, the design point for the elementary event of excursion at

time tk of the response of a linear structure to a Gaussian excitation can be found

in closed form. Assuming for the time being and without loss of generality that the

Gaussian excitation is a stationary white noise random process W(t) of intensityS0,

its discretization in terms of random pulses is carried out as usual according to

Ys=

ts+tts

W(t) dt= ys

2S0t

Direct constraint minimization to find the minimum distance point on the limit-state

surface can be carried out analytically for linear systems using Lagrange multipliers.

The functional whose stationarity yields the solution is

f(y) = y2 +G (y) =Ns=1

y2s+

bk

kNs=1

ys

2S0t

h (k s+ 1)

where y2 is the quantity to be minimized, is the, yet to be determined, Lagrangemultiplier and G(y) = 0 is the constraint. The latter has a form equivalent to that

given in Eq.(2.3-3) but using summation in place of the dot product. The vectorhis


48/147


the discrete counterpart of the system unit-impulse response function h and collects

values ofh at discrete time tk = (k

1)t. Stationarity conditions are obtained

setting to zero the partial derivatives of the functional above with respect to the

components of the solution y and the multiplier

f(y)

ys= 2ys

2S0t h (k s+ 1) H (k s) = 0

f(y)

=bk

kNs=1

ys

2S0t

h (k s+ 1) = 0

where H(k s) is the Heaviside unit-step function. Solving the first of the twoequations with respect to ys one gets

ys=

2

2S0t h (k s+ 1) H (k s)

Substitution ofys in the second equation yields the value of the Lagrange mul-

tiplier

= 2bk

2S0tkNs=1

h2 (k s+ 1)=

2bk2k

in which 2k is the variance of the linear response at time step k

2k = E u2 (k)= E k

s=1

Ysh (k

s+ 1)

2

== E

ks=1

Ysh (k s+ 1)k

q=1

Yqh (k q+ 1)

= 2S0tk

s=1

h2 (k s+ 1)

due to the independence of the acceleration pulses. Finally, the s-th component of

the design point is obtained in the form


49/147


ys

= bk

2k2S0t h (k s + 1) H (k s)

Apart from a constant factor depending on the threshold bk at time tk and on the

intensity of the white noise base excitation S0, one sees that the design point is, for

ts tk, proportional to the mirror-image of the unit-impulse response function ofthe system6, and zero afterward as implied by the Heaviside step function.

The discrete counterpart of the corresponding input base acceleration process is

simply proportional to the design point

Y (s) =Yst

=

2S0

t ys

In Figure(2-6) the displacement unit-impulse response function h of a linear

visco-elastic system with period T = 1sec and damping ratio = 0.10 is plotted

together with the non-stationary variance of this system responseu(t) to a stationary

white noise excitation of intensity S0= 1m2/sec3. Also plotted are the design point

excitation W(t) for a threshold b = 0.5mat time tk = 8sec and the corresponding

design point response. The response reaches the assigned threshold for the firsttime at tk = 8sec and having an horizontal tangent, i.e. with zero kinetic energy.

The latter condition implies that the state of the system is one of minimum for

the potential energy among all those for which the threshold is attained. This

is one more way to see that this solution is optimal: the optimality requirement

in terms of potential energy (minimize potential energy given that response has

reached the threshold at time tk) has been shown to be equivalent to that in terms

of displacement in [40].

It is again observed that for any threshold and excitation the design point atany time instant is known analytically, i.e. without the necessity of performing

any numerical optimization, once the system unit-impulse response is known. This

is true also for the more general non-white non-stationary excitations presented

before, as for example the amplitude modulated filtered white-noise processes, once

6In fact for s1 < s2 one hash (k s1+ 1) < h (k s2+ 1)


50/147


0 5 100.2

0.15

0.1

0.05

0

0.05

0.1

0.15

t [sec]

h[m]

(a)

0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t [sec]

[m]

(b)

0 5 106

4

2

0

2

4

6

8

t [sec]

W* [m/sec

2]

(c)

0 5 100.4

0.2

0

0.2

0.4

0.6

t [sec]

u* [m]

(d)

Figure 2-6: Design point for a linear system under stationary Gaussian white noise

excitation: (a) unit-impulse response function; (b) variance of the response; (c)design-point excitation; (d) design-point response.

the system is redefined to include the dynamics of both the filter and the structure

and the envelope function is taken into account.

Considering again Figure(2-6(c)) one observes that the design point excitation

is, as anticipated, a nearly harmonic function of time with gradually increasing

amplitude and frequency that, in this case, coincides with that of the system. It

compares oddly with the sample of the stationary white noise shown in Figure(2-

7), but the difference is understood once it is recognized that, while this ground

acceleration corresponds to the minimum ofy conditionalony lying on the limit-state surface, the acceleration sample in Figure(2-7) is totally random and, for this


51/147


reason, the comparison is not a proper one.

The reliability index corresponding to the design point evolves with time inverselywith the response standard deviation as shown by

k = y = N

s=1

(ys)2 =

ks=1

(ys)2 =

bkk

and is plotted in Figure(2-8) for the same system and excitation of Figure(2-6). One

can see that the distance of the hyperplane from the origin decreases quickly with

time and becomes constant when the response attains stationarity.

0 2 4 6 8 10100

80

60

40

20

0

20

40

60

80

100

t [sec]

W(t)

Figure 2-7: White noise sample.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t [sec]

Figure 2-8: Reliability index evolutionwith time.

2.3.3 From the elementary excursion events to the first ex-cursion event: non-linear systems and mean out-crossingrate

Having studied the elementary excursion event one can now focus on the more

complex first excursion event and on the computation of the associated probability.

Since failure at any time tk implies first-excursion failure, the failure domainFof this event is the union of the elementary domains Fk or, in the terminology


52/147


of systems theory, the component events are arranged in series to give the first

excursion one. This can be written as

F=Nk=1

Fk =Nk=1

{bk u (tk) 0}

where for linear systems Fk is thek-th half-space having normalk and distancekfrom the origin.

For non linear systems Fk, the boundary ofFk, is non linear and numericaloptimization is necessary. Performing FORM at each time step tk is clearly not

an option. In this case an approximate solution is offered by the mean crossing

rate k of excursion above level bk. In fact, assuming that the system starts in the

safe domain7, the probability of failure for the first excursion event in time D is

the complement of the probability of having zero crossings within D, hence we can

write, denoting by N(D) the number of crossings in D,

Pf= 1

Pr

{N(D) = 0

}=

n=1

Pr

{N(D) =n

}

n=1

n Pr {N(D) =n} = E [N(D)] = D0

(t) dt (2.3-5)

The right-most integral in the inequality above constitutes an upper bound to

Pfand can be computed by a small number of evaluations of the integrand function

(t), since the mean crossing rate variation is much slower than that of the process

itself. It is clear that the quality of the bound in Eq.(2.3-5) strongly depends on

how large is bk with respect to the response standard deviation k and that it canbe reliably used only when the probability of having two or more crossings in dt is

negligible with respect to Pr{N(D) = 1}.What is now needed is a way to compute the mean crossing rate inside the

framework that has been presented in this chapter, i.e. for processes reduced to

7The point y0 representing the system and the loads at time t0 = 0 belongs to the safe domain.


53/147


vectors of time-invariant random variables . It is now appropriate to observe that,

beside the load variables that come from the earthquake process discretisation,

the vector can include additional non-load variables chosen to represent random

mechanical properties of the analysed structure. One can imagine that also these

additional variables come from discretisation of random fields describing material

properties variation over the structural domain.

If g (t, ) = g [u (t, )] denotes a time-dependent scalar limit-state function de-

fined in the physical space of the basic random variables as a function of the

response process u (t, ) in the usual way such as g > 0 defines the set of safe

values of and g 0 the corresponding set of failure values, one can computethe bound given in Eq.(2.3-5) using the mean crossing rate g(t) of g(t) below the

threshold g= 0.

Starting from the original derivation of Rice formula for the crossing rate [54], it

is easily seen that, essentially, the probability of having a crossing at time t is given

by the probability that the limit-state function is above zero at time t and below it

at time t+t, infinitely close to t. Hence the mean crossing ratecan be computed

as [36]

(t) = limt0

P r {g1() 0 g2() 0}t

(2.3-6)

where the two time-independent limit state functions are defined as follows

g1() = g(u(, t)) (2.3-7)g2() = g(u(, t+t)) = g(u(, t)) + ug(u(, t))u(, t)t (2.3-8)

in terms of the time-dependent limit-state function g. For the solution of Eq.(2.3-6),

one might think of using directly as random variablesu(t) andu(t + t) oru(t) and

u(t) as for the second equality in Eq.(2.3-8). This would be unpractical, or even

unfeasible, due to the difficulty of determining the joint PDF of the two functions8.

Since this latter probability distribution is needed for the transformation from the

original to the standard normal space, it is in any case convenient to work in terms

8It is recalled that only for stationary Gaussian response the displacement and the velocityprocesses are independent.


54/147


of the basic variables , even if this implies in general a considerable increase in the

problem dimension given that may include several hundreds of random pulses ys,

in addition to the subset of mechanical parameters.

The event whose probability appears in the numerator of Eq.(2.3-6) is a par-

allel event [36], the intersection of the component events. A first order (FORM)

approximation to the probability of the intersection is given by

Pf=2(1, 2, 12) = (1) (2) + 120

2(1, 2, ) d (2.3-9)

where 2() and () are the standard normal bivariate PDF and univariate CDFrespectively,i is the reliability index for limit state function Gi(y) =gi((y))

9 and

the correlation coefficient between the two linearized events G1 = 0 and G2 = 0

is given by the dot product of their alpha vectors 12 = T12, see Appendix

SectionA.1.3.3. In Figure (2-9) the two limit-state surfaces and their linear ap-

proximation at their respective design points are shown in the standard normal

space. It is noted that the computation of the bivariate (and, more generally, of the

n-variate) Gaussian CDF is numerically difficult, especially when the correlationcoefficient between the events is close to 1 as in this case. A number of algorithmsare available to perform the integral (2.3-9), see for example [1].

It is recalled that the computationally most expensive part of the FORM analysis

is the search for the design point. Efficient algorithms to perform this task require

computation of the gradient of the limit-state function in the basic variables space.

Computation of this gradient by finite differences is prohibitive and hence a direct

approach is necessary to optimize computation. Postponing a detailed discussion

of this issue to the next chapter, only the expressions of the gradients ofG1 and

G2 are given here, which are easily obtained by making use of the chain rule of

differentiation

9Being the design point defined as the point of minimum distance in the standard normal space,it is in this latter space that the search procedure is carried out.


55/147


Figure 2-9: Crossing rate as parallel event (FORM approximation).

yG1=

g1Jy=

ugJuJy (2.3-10)

yG2 = g2Jy= (ugJu+ u(ug) ut + ugJut) Jy (2.3-11)

where Jrs are jacobian matrices of derivatives ri/sj .

It is recalled that the vector is not limited to those load random variables

that come from the discretisation of the input random process but can include

any number and kind of random variables chosen to describe randomness in the

mechanical parameters governing both the dynamic response of the structure and

the capacity of the considered failure mechanisms. This is an important feature

of this FORM-based approximate solution of the first-excursion problem, as it is

the fact that inelastic material behavio

Franch in Thesis

Documents

Transcript of Franch in Thesis