PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
P-Wave Onset Point Detection for Seismic Signal Using Bhattacharyya DistanceCSCJournals
In seismology Primary p-wave arrival identification is a fundamental problem for the geologist worldwide. Several numbers of algorithms that deal with p-wave onset detection and identification have already been proposed. Accurate p- wave picking is required for earthquake early warning system and determination of epicenter location etc. In this paper we have proposed a novel algorithm for p-wave detection using Bhattacharyya distance for seismic signals. In our study we have taken 50 numbers of real seismic signals (generated by earthquake) recorded by K-NET (Kyoshin network), Japan. Our results show maximum standard deviation of 1.76 sample from true picks which gives better accuracy with respect to ratio test method.
Spectral estimation, and corresponding time-frequency representation for nonstationary signals, is a cornerstone in geophysical signal processing and interpretation. The last 10–15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. These conventional techniques, like the short-time Fourier transform and the continuous wavelet transform, show some limitations in terms of resolution (localization) due to the trade-off between time and frequency localizations and smearing due to the finite size of the time series of their template. Well-known techniques, like autoregressive methods and basis pursuit, and recently developed techniques, such as empirical mode decomposition and the synchrosqueezing transform, can achieve higher time-frequency localization due to reduced spectral smearing and leakage. We first review the theory of various established and novel techniques, pointing out their assumptions, adaptability, and expected time-frequency localization. We illustrate their performances on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event, and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. Finally, their outcomes are discussed and possible avenues for improvements are proposed.
P-Wave Onset Point Detection for Seismic Signal Using Bhattacharyya DistanceCSCJournals
In seismology Primary p-wave arrival identification is a fundamental problem for the geologist worldwide. Several numbers of algorithms that deal with p-wave onset detection and identification have already been proposed. Accurate p- wave picking is required for earthquake early warning system and determination of epicenter location etc. In this paper we have proposed a novel algorithm for p-wave detection using Bhattacharyya distance for seismic signals. In our study we have taken 50 numbers of real seismic signals (generated by earthquake) recorded by K-NET (Kyoshin network), Japan. Our results show maximum standard deviation of 1.76 sample from true picks which gives better accuracy with respect to ratio test method.
Spectral estimation, and corresponding time-frequency representation for nonstationary signals, is a cornerstone in geophysical signal processing and interpretation. The last 10–15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. These conventional techniques, like the short-time Fourier transform and the continuous wavelet transform, show some limitations in terms of resolution (localization) due to the trade-off between time and frequency localizations and smearing due to the finite size of the time series of their template. Well-known techniques, like autoregressive methods and basis pursuit, and recently developed techniques, such as empirical mode decomposition and the synchrosqueezing transform, can achieve higher time-frequency localization due to reduced spectral smearing and leakage. We first review the theory of various established and novel techniques, pointing out their assumptions, adaptability, and expected time-frequency localization. We illustrate their performances on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event, and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. Finally, their outcomes are discussed and possible avenues for improvements are proposed.
Robust SINS/GNSS Integration Method for High Dynamic ApplicationsRadita Apriana
As high dynamic movement is always accompanied by colored noise which lacks of mathematical
model, traditional Kalman filtering based on an assumption of white Gaussian noise always faces serious
divergence. To enhance the performance in high dynamic environment with uncertain colored noise, a kind
of robust filtering based on H-infinity technology is developed. State model of the algorithm is derived from
SINS error propagation. Both position and velocity errors are used as the measurements. A simulation
system which includes a tri-axial turntable and a GNSS signal simulator is used to verify the integration
design under high dynamic environment. Simulation results proved that both the accuracy and robustness
of the integration design have been improved significantly.
The stochastic system is very importment in many aspacts. Wiener processes is a sort of importment
stochastic processes. Wiener square processes is a class of useful stochastic processes in practies,its study is very
value.In this paper,we study Wiener square processes using haar wavelet and wavelet transform.we study its some
properties and wavelet expansion. Index Wiener Integral processes is a class of useful stochastic processes in
practies,its study is very value.In this paper,we study it using haar wavelet and wavelet transform on [0,t].we study
its some properties and wavelet expansion
Accuracy of the internal multiple prediction when a time-saving method based ...Arthur Weglein
The inverse scattering series (ISS) is a direct inversion method for a multidimensional acoustic,
elastic and anelastic earth. It communicates that all inversion processing goals can be
achieved directly and without any subsurface information. This task is reached through a taskspecific
subseries of the ISS. Using primaries in the data as subevents of the first-order internal
multiples, the leading-order attenuator can predict the time of all the first-order internal multiples
and is able to attenuate them.
GPR Probing of Smoothly Layered Subsurface Medium: 3D Analytical ModelLeonid Krinitsky
An analytical approach to GPR probing of a
horizontally layered subsurface medium is developed, based on the coupled-wave WKB approximation. An empirical model of current in dipole transmitter antenna is used.
Calcolo della precompressione:
DOMINI e STRAUS7
Corso di Gestione di Ponti e Grandi Strutture A.A. 2021/22
Prof. Ing. Franco Bontempi
Facoltà di Ingegneria Civile e Industriale
Sapienza Università di Roma
Scopo dell'evento è
• illustrare l'identità culturale, e tecnica – di cui il progetto è parte fondante – del SSD Tecnica delle Costruzioni nella didattica,
• evidenziando contemporaneamente le opportunità di collaborazione trasversale con altre discipline,
• con particolare riferimento ai corsi della lauree magistrali o
equivalenti, e livelli di formazione successivi (master e dottorati).
L’incontro ha l’obiettivo di delineare l'identità culturale, scientifica e tecnica della disciplina della Tecnica delle Costruzioni nella didattica, evidenziando contemporaneamente le opportunità di collaborazione trasversale con altre discipline, con particolare riferimento ai corsi della lauree magistrali o equivalenti, e livelli di formazione successivi (master e dottorati).
In recent years, there has been an increasing interest in permanent observation of the dynamic behaviour of bridges for longterm
monitoring purpose. This is due not only to the ageing of a lot of structures, but also for dealing with the increasing
complexity of new bridges. The long-term monitoring of bridges produces a huge quantity of data that need to be effectively
processed. For this purpose, there has been a growing interest on the application of soft computing methods. In particular,
this work deals with the applicability of Bayesian neural networks for the identification of damage of a cable-stayed bridge.
The selected structure is a real bridge proposed as benchmark problem by the Asian-Pacific Network of Centers for Research
in Smart Structure Technology (ANCRiSST). They shared data coming from the long-term monitoring of the bridge with the
structural health monitoring community in order to assess the current progress on damage detection and identification
methods with a full-scale example. The data set includes vibration data before and after the bridge was damaged, so they are
useful for testing new approaches for damage detection. In the first part of the paper, the Bayesian neural network model is
discussed; then in the second part, a Bayesian neural network procedure for damage detection has been tested. The proposed
method is able to detect anomalies on the behaviour of the structure, which can be related to the presence of damage. In order
to obtain a confirmation of the obtained results, in the last part of the paper, they are compared with those obtained by using a
traditional approach for vibration-based structural identification.
In recent years, structural integrity monitoring has become increasingly important in structural engineering and construction management. It represents an important tool for the assessment of the dependability of existing complex structural systems as it integrates, in a unified perspective, advanced engineering analyses and experimental data processing. In the first part of this work
the concepts of dependability and structural integrity are
discussed and it is shown that an effective integrity assessment
needs advanced computational methods. For this purpose, soft computing methods have shown to be very useful. In particular, in this work the neural networks model is chosen and successfully improved by applying the Bayesian inference at four hierarchical levels: for training, optimization of the regularization terms, databased model selection, and evaluation of the relative importance of different inputs. In the second part of the article,
Bayesian neural networks are used to formulate a
multilevel strategy for the monitoring of the integrity of long span bridges subjected to environmental actions: in a first level the occurrence of damage is detected; in a following level the specific damaged element is recognized and the intensity of damage is quantified.
This paper deals with the general framework for the development and the maintenance of complex structural systems. In the first part, starting with a semantic analysis of the term ‘structure’, the traditional approach to structural problem solving has been reconsidered. Consequently, a systemic approach for the formulation of the different kinds of direct and inverse problems has been framed, particularly with regards to structural design and
maintenance. The overall design phase is defined with the aid of the performance-based design (PBD) philosophy, emphasizing the concepts of dependability and enlightening the role of structural identification. The second part of the present work analyses structural health monitoring (SHM) in the systemic way previously introduced. Finally, the techniques related to the implementation of the monitoring process are introduced and a synoptic overview of methods and instruments for structural health monitoring is
presented, with particular attention to the ones necessary for structural damage identification.
Disegni strutturali e particolari costruttivi di ponti in cemento armato raccolti dall'Ing. Cosimo Bianchi.
Ad uso esclusivo degli Allievi del Corso di Teoria e Progetto di Ponti della Facoltà di Ingegneria della Sapienza - Prof. Ing. Franco Bontempi
Disegni strutturali e particolari costruttivi di ponti in acciaio raccolti dall'Ing. Cosimo Bianchi.
Ad uso esclusivo degli Allievi del Corso di Teoria e Progetto di Ponti della Facoltà di Ingegneria della Sapienza - Prof. Ing. Franco Bontempi
Libro che raccoglie le lezioni del Prof. Giulio Ceradini a cura del Prof. Carlo Gavarini.
Ad uso esclusivo degli Allievi del Corso di Teoria e Progetto di Ponti della Facoltà di Ingegneria della Sapienza - Prof. Ing. Franco Bontempi
A numerical approach to the reliability analysis of reinforced and prestressed concrete structures is presented. The problem is formulated in terms of the probabilistic safety factor and the structural reliability is evaluated by Monte
Carlo simulation. The cumulative distribution of the safety factor associated with each limit state is derived and a reliability index is evaluated. The proposed procedure is applied to reliability analysis of an existing prestressed concrete arch bridge.
This paper presents a general approach to the probabilistic prediction of the structural service life and to the maintenance
planning of deteriorating concrete structures. The proposed formulation is based on a novel methodology for the assessment of the time-variant structural performance under the diffusive attack of external aggressive agents. Based on this methodology, Monte Carlo
simulation is used to account for the randomness of the main structural parameters, including material properties, geometrical parameters, area and location of the reinforcement, material diffusivity and damage rates. The time-variant reliability is then computed with respect to proper measures of structural performance. The results of the lifetime durability analysis are finally used to select, among different maintenance scenarios, the most economical rehabilitation strategy leading to a prescribed target value of the structural service life. Two numerical applications, a box-girder bridge deck and a pier of an existing bridge, show the effectiveness of the proposed methodology.
This paper presents a novel approach to the problem of durability analysis and lifetime assessment of concrete structures under
the diffusive attack from external aggressive agents. The proposed formulation mainly refers to beams and frames, but it can be easily
extended also to other types of structures. The diffusion process is modeled by using cellular automata. The mechanical damage coupled to diffusion is evaluated by introducing suitable material degradation laws. Since the rate of mass diffusion usually depends on the stress state, the interaction between the diffusion process and the mechanical behavior of the damaged structure is also taken into account by a proper modeling of the stochastic effects in the mass transfer. To this aim, the nonlinear structural analyses during time are performed
within the framework of the finite element method by means of a deteriorating reinforced concrete beam element. The effectiveness of the
proposed methodology in handling complex geometrical and mechanical boundary conditions is demonstrated through some applications.
Firstly, a reinforced concrete box girder cross section is considered and the damaging process is described by the corresponding evolution of both bending moment–curvature diagrams and axial force-bending moment resistance domains. Secondly, the durability analysis of a
reinforced concrete continuous T-beam is developed. Finally, the proposed approach is applied to the analysis of an existing arch bridge and to the identification of its critical members.
The paper deals with the assessment during time of r.c. structures under damage due to diffusion of external agents inside the structure. The diffusion process is modelled by a cellular automata based approach, taking the interaction with the mechanical state of the structures, i.e. the cracking state of the structures, into account. A so-called staggered process then solves the coupled problem. An application shows the effectiveness of the proposed analysis strategy, together some design considerations about the structural robustness.
Atti Congresso CTE, Pisa 2000
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Design and Analysis of Algorithms-DP,Backtracking,Graphs,B&B
Lecture 2 sapienza 2017
1. Academic excellence for business and the professions
Lecture 2:
Linear random vibrations analysis for
seismically excited structures
Lecture series on
Stochastic dynamics and Monte Carlo simulation
in earthquake engineering applications
Sapienza University of Rome, 13 July 2017
Dr Agathoklis Giaralis
Visiting Professor for Research, Sapienza University of Rome
Senior Lecturer (Associate Professor) in Structural Engineering,
City, University of London
2. 2
2
0
0
2
I D S
t
eff g
n n g
f t f t f t
mu t cu t ku t
mu t cu t ku t p t mu t
u t u t u t u t
Linear damped SDOF system under ground excitation
(a reminder)
2 2
2 2t t t
n n n g n gu t u t u t u t u t Alternatively:
3. Linear damped single degree of freedom oscillator with zero initial conditions
For we get: For we get:
A more concise and efficient approach to exploit the above results requires
complex numbers analysis
Consider
We seek a harmonic/steady state solution of the form:
It can be readily shown (e.g. Chopra 2001):
And (Dynamic amplification factor):
22 2
1
1 / 2 /st
n n
H
u
22
1 1/
1 / 2 /n n
k
H
m i c k i
1/stu k
with:
Steady-state response to harmonic excitation
(a reminder)
4. Assume harmonic ground displacement:
4
gi t
gu t e
2 gi t
eff gp t m e
Then: And:
2
22 2
/
1 / 2 /
g n
g
g n g n
u t
u t
22 22
1
1 / 2 /g
n g n g n
u t
H
u t
𝜔 𝑔 𝜔 𝑛𝜔 𝑔 𝜔 𝑛
Steady-state response to harmonic excitation
(a reminder)
5. 5
Any periodic or finite duration function with period
can be written as
with:
OR:
with:
0
1
i j t i j t i j t
j j j
j j
p t P P e P e P e
0 0P a
2
j j
j
a ib
P
*
2
j j
j j
a ib
P P
0
oT i j t
jP p t e dt
0 0
1 1
2 2
cos sin cos sinj j j o j o
j jo o
j j
p t a a t b t a a j t b j t
T T
p t oT
2 / oT
0 0 0 0
1 2 2
; cos ; sin
o o oT T T
j j
o o o
a p t dt a p t j t dt b p t j t dt
T T T
1i where:
Response to arbitrary deterministic excitations
(SDOF systems)
6. where
it can be readily seen that: where
So:
Response to arbitrary deterministic excitations
(SDOF systems)
For aperiodic excitations:
7. Convolution theorem is very general!
Impulse response function
Response by Duhamel’s (convolution) integral
Frequency response function
Response by a simple multiplication
in the frequency domain
2
1/
1 / 2 /n n
k
i
Response to arbitrary deterministic excitations
(SDOF systems)
8.
22 2
1
1 / 2 /st
n n
H
u
Normalised amplification factor
Linear SDOF systems act like “pass-band” filters!
10. Soil acts like a dynamic oscillator and greatly affects the ground motions that a structure
atop the soil column experiences.
Example: Local soil effects (to be revisited later… the Kanai-Tajimi!)
Rock
Equivalent to
Softer and deeper soils will have longer predominant frequency content in terms of
natural period
Certain “softer” soils may also magnify the ground motion
Linear SDOF systems act like “pass-band” filters
and relation to “resonance”
11. Linear SDOF systems under random excitation
Mean value of the response
It holds:
Ensemble averaging:
Change sequence of integration:
Stationarity: or:
Recall
12. Linear SDOF systems under random excitation
Input/output autocorrelation relationship
It holds:
Ensemble averaging:
Change sequence of integration and noting (from stationarity)
we get:
13. Linear SDOF systems under random excitation
Input/output power spectral density relationship
It holds:
and we add 3 exponential terms whose product is “1”:
Therefore:
14. Linear SDOF systems under random excitation
Mean square response (== variance for zero-mean processes)
From the autocorrelation function:
Example: white input noise
From the PSD:
15. Linear SDOF systems under random excitation
A few more important results (e.g. Newland 1993)
2 2
{ ( )} { ( )} { ( )} 0
1 1
{ ( ) ( )} { ( )} { ( )} 0
2 2
d d
E x t E x t E x t
dt dt
d d
E x t x t E x t E x t
dt dt
Derivatives of stationary stochastic
process (e.g. velocity)
are zero due to time-independence
BUT (for cross-correlations and cross-power spectral densities):
2
( ) { ( ) ( )} [ { ( ) ( )}] ( )
( ) { ( ) ( )} ( )
: ( ) ( ) ( ) ( )
xx xx
xx xx
xx xx xx xx
d d
R t t t t R
d d
d
R t t R
d
and also S i S S S
which means (for example)
2
( ) ( )yy xxS H S
16. Linear SDOF systems under random excitation
Response to (ideal) white noise (e.g. Crandall and Mark 1963)
Auto-correlation function Power spectrum
== Square magnitude of the FRF (input
base acceleration output displacement)
of a SDOF (times So)
2
3
2
o
y
n
S
Variance of displacement:
Variance of velocity:
2
2
o
y
n
S
17. Linear SDOF systems under random excitation
Response to band-limited white noise White noise is an “approximation”
18.
2
2
22 2
2
1 4
1 4
g
g
KT
g
g g
S
Kanai-Tajimi (KT case)
(Kanai, 1957):
Introduces only two more unknowns (ζg, ωg) in the
optimization problem.
Has strong and clear soil characterization
capabilities: ζg and ωg can be interpreted damping
ratio and natural frequency of the surface soil layers.
Does not suppress the low frequency energy of the
process.
Rock
Equivalent
to
2
2
2
2
t t t
n n
n g n g
u t u t u t
u t u t
Commonly used power spectra to model seismic processes
Input: white noise acceleration
output: total displacement
19. Commonly used power spectra to model seismic processes
2
22 2
2
1 4
f
CP KT
f
f f
S S
Filters the low frequency energy by a second order
high-pass (H-P) filter
Further increase of the number of parameters to be
defined (ζf, ωf , ζg, ωg).
The values attained by the parameters
corresponding to the bedrock can be unrealistic.
Clough-Penzien (CP case)
(Clough and Penzien, 1975):
Rock
Equivalent
to
Additional filter with little
physical intuition….
20. Commonly used power spectra to model seismic processes
Butterworth filtered Kanai-
Tajimi
2
2 2
N
BWKT KTN N
o
S S
The low frequency content introduced by the Kanai-Tajimi part of
the spectrum can be filtered out of the process more effectively than
in the CP case.
The assets of the Kanai-Tajimi spectral form are maintained; only
two parameters need to be defined which unambiguously reflect on
the soil conditions associated with the form of the design spectrum.
The order (N) and the cut-off frequency (ωo) of the H-
P filter can be judicially selected, so that:
Giaralis and Spanos (2009)
21. Giaralis and Spanos (2009)
Commonly used power spectra to model seismic processes
Manifestation of singularity of the unfiltered Kanai-Tajimi spectrum
23. Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
2
2 ; 0 0 0n n gy t y t y t u t y y
cos sinn n ny t a t t t and y t a t t t
Assuming that the input process is relatively broadband compared to the transfer
function of the oscillator (ζ<<1), the relative displacement y(t) of the oscillator is
well approximated by the process:
This is a “Pseudo-harmonic” response and the
envelop represents well local peak responses.
Expected frequency in terms of
moments of the PSD (Rice 1946)
24. Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
For narrow-band processes
It can be proved that it follows a Rayleigh
distribution for Gaussian input processes
25. We are after “positive” up-crossings of a threshold α
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
We focus on the statistics of να
+ ->
26. Core equation relating a Sd for Gaussian input stochastic process of finite
duration Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
The peak factor ηj is the constant by which
the standard deviation of the response of a
linear SDOF oscillator must be multiplied
to predict the level Sd below which the
peak response of the considered oscillator
will remain, with probability p, throughout
the duration of the input process Ts.
,0,d j j GS
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
where
2
2 22 2
1
,
2
s
j j j
H T
1 exp 2
j
j sT
and
2
, ,, ,
0
; ,m
s j m G sj m G
T G H T d
27. where
The response spectral moments are given as (Vanmarcke 1976)
,0,d j j GS
2
2 22 2
1
,
2
s
j j j
H T
1 exp 2
j
j sT
and
2
, ,, ,
0
; ,m
s j m G sj m G
T G H T d
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
28. Core equation relating a Sd for Gaussian stochastic process of finite duration
Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
and
2*
1,2, ,1,
,0, ,0, ,2,
ln ; 1
2
j G j Gs
j j
j G j G j G
T
v p q
,0,*
,0,
exp 2 1
/ 2
s j G
s s
s j G
T
T T
T
where
1.2
; 2ln 2 1 exp ln 2j j j jv q v
,0,d j j GS
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
2
G H
29. Definition of the response spectrum
, maxd n
t
S x t Displacement response spectrum:
, ,v n n d nS S Pseudo-velocity response spectrum:
Pseudo-acceleration response spectrum: 2
, ,a n n d nS S
Kramer, 1996
30. Core equation relating a Sd for Gaussian input stochastic process of finite
duration Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
2
,0,
2
,a j j j G
j
S
and
2*
1,2, ,1,
,0, ,0, ,2,
ln ; 1
2
j G j Gs
j j
j G j G j G
T
v p q
,0,*
,0,
exp 2 1
/ 2
s j G
s s
s j G
T
T T
T
where
1.2
; 2ln 2 1 exp ln 2j j j jv q v
The peak factor ηj is the constant by which
the standard deviation of the response of a
linear SDOF oscillator must be multiplied
to predict the level Sd below which the
peak response of the considered oscillator
will remain, with probability p, throughout
the duration of the input process Ts.
Reponse spectrum compatible stationary processes
31. where ω0 is such that
2 1
02
11
0
2 / ,4
;
4
0 ; 0
k
kk
i k N
ik k k k k
k
S
G
G
Approximate numerical scheme to recursively evaluate G(ω) at a specific set
of equally spaced by Δω (in rad/sec) natural frequencies ωj= ω0+ (j-0.5)Δω; j=
1,2,…,N (Cacciola et al., 2004; Giaralis and Spanos, 2010)
1.2
0 min ln 2 1 exp ln 2 0
k
k k kv q v
2 ln 0.5
s
k k
T
v
2
1
2 2
1 2
1 1 tan
1 1
kq
and the peak factors can be computed by assuming stationary white noise
input (Der Kiureghian 1980)
1.2
2ln 2 1 exp ln 2k k k kv q v
andwhere
Reponse spectrum compatible stationary processes
32. Reponse spectrum compatible stationary processes
2
1
,
,
a jv v
j j v
j
S
G G
A
Iterative modification of the obtained discrete power spectrum G[ωj] to
improve the matching of the associated response spectrum A[ωj,ζ] with the
target design spectrum Sα (e.g. Sundararajan, 1980; Gupta and Trifunac, 1998)
2
,0,
2
, j j j G
j
A
2*
1,2, ,1, ,0,*
,0, ,0, ,2, ,0,
ln ; 1 ; exp 2 1
2 / 2
sj G j G j Gs
j j s s
j G j G j G s j G
TT
v p q T T
T
1.2
; 2ln 2 1 exp ln 2j j j jv q v
where
Moments can be numerically computed very efficiently (Di Paola & La Mendola 1992)