Earthquake
Response
of Inelastic
Systems
Outline
Giacomo Bo
Motivation
Earthquake Response
of Inelastic Systems
Giacomo Bo
Cyclic Behaviour
E-P Idealisation
Earthquake
Response of E-P
Systems
Eects of Yielding
Earthquake
Response
of Inelastic
Systems
Giac
Earthquake
excitation
Outline
Earthquake
excitation
Giacomo Bo
Earthquakes and
Earthquake
Response
Earthquake excitation
Giacomo Bo
Earthquakes and
Earthquake
Response
Response
Spectrum
Response
Spectrum
Characteristics
Giacomo Bo
Dipartimento di Ingegner
Dynamics of Structures
First Home Assignment 2013-14
due by May 20th 2014
Giacomo Bo
May 6, 2014
Instructions
This assignment is due on Tuesday May 20th. At your choice, you can either mail
me your paper, in the form of a single PDF document, or hand it i
The second 2014 Dynamics of Structures home work
due on the day of your oral exam
Instructions
You shall present your work in the form of a paper, either printed or nicely handwritten, complete of all the required gures1 and a printed listing of all the
s
Dynamics of Structures 2009-2010
summer home assignment
1
Dynamical Testing
A simply supported beam is loaded at mid-span with a harmonic load, with
dierent frequencies and the same nominal load amplitude.
Due to the position of the load and the frequenci
SDOF linear
oscillator
Outline
SDOF linear
oscillator
Giacomo Bo
Response to
Periodic Loading
SDOF linear oscillator
Response to Periodic and Non-periodic Loadings
Giacomo Bo
Response to
Periodic Loading
Fourier Transform
Fourier Transform
The Discrete
Fo
Structural
Matrices
Outline
Structural
Matrices
Giacomo Bo
Introductory
Remarks
Structural Matrices in MDOF Systems
Giacomo Bo
Structural
Matrices
Evaluation of
Structural
Matrices
Choice of Property
Formulation
Dipartimento di Ingegneria Civile e Ambient
SDOF linear
oscillator
Giacomo Bo
SDOF linear oscillator
Response to Harmonic Loading
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
March 16, 2011
Outline of parts 1 and 2
SDOF linear
oscillator
Giacomo Bo
Response of an Undampe
Dynamics of Structures 2011-2012
Second Home Assignment, due on your oral exam
The day of your oral examination you shall submit to me a printed or handwritten
paper with your solutions to the following three problems.
1
Dierential support motion
x2
m
x1
SDOF linear
oscillator
Giacomo Bo
SDOF linear oscillator
Response to Periodic and Non-periodic Loadings
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
March 27, 2012
Outline
SDOF linear
oscillator
Giacomo Bo
Response to Periodic
SDOF linear
oscillator
Outline
SDOF linear
oscillator
Giacomo Bo
Response to
Impulsive Loading
SDOF linear oscillator
Response to Impulsive Loads & Step by Step Methods
Giacomo Bo
Response to
Impulsive Loading
Review
Review
Step-by-step
Methods
Examples o
SbS Methods,
Rigid Bodies
Giacomo Bo
Examples of SbS
Methods
Numerical Integration Rigid Assemblages
Step-by-step Numerical Procedures
Introduction to Complex Systems
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
April 12, 2012
Superposition
Eigenvector Expansion
Superposition
Giacomo Bo
Giacomo Bo
Eigenvector
Expansion
Response by Superposition
Uncoupled
Equations of
Motion
Eigenvector
Expansion
For a N-DOF system, it is possible and often advantageous
to represent the displace
Truncation Errors,
Correction
Procedures
Giacomo Bo
Rayleigh-Ritz
Example
Subspace iteration
Truncation Errors, Correction Procedures
How many
eigenvectors?
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
May 23, 2012
Truncation E
Continuous
Systems
Problem statement
Continuous
Systems
Giacomo Bo
Giacomo Bo
Problem
statement
Problem
statement
Solution
Continuous Systems
+
P
vt
Solution
u(x, t)
an example
x
L
Giacomo Bo
A uniform beam (m(x) = m, EJ(x) = EJ) of lenght L is
loaded by
Generalized
SDOFs
Giacomo Bo
Introductory
Remarks
The
Homogeneous
Problem
Multi Degrees of Freedom Systems
MDOFs
Modal Analysis
Examples
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
May 4, 2012
Outline
Generalized
SDOFs
Introdu
New First Assignment
due on September 1st
Giacomo Bo
Instructions
Due to logistical issues you must submit this homework by email only, in the form of a PDF
attachment produced by a word processor or a typesetting system (absolutely no scans of your
handw
Free Vibrations of an E-P SDOF
Giacomo Bof
April 1, 2014
%pylab inline
In [1]: %config InlineBackend.figure_format = svg
figsize(8,1.5)
Populating the interactive namespace from numpy and matplotlib
0.1 Elasto-plastic response
In our SDOF system we have a
Aprogramtopreparethemassmodalexpansionfunctionstobeplotted
withthehelpofsmallscript
Program:
from scipy import *
from scipy.optimize import newton
from scipy.integrate import quad
to8=range(8)
betas=[newton(lambda x: cos(x)+1/cosh(x), (2*n+1)*pi/2) for n
Generalized
SDOFs
Outline
Generalized
SDOFs
Giacomo Bo
Giacomo Bo
Continuous
Systems
Generalized Single Degree of Freedom
Systems
Continuous
Systems
Vibration Analysis
by Rayleighs
Method
Vibration Analysis
by Rayleighs
Method
Renement of
Rayleighs
Estima
Dynamics of Structures 2009-2010
1st home assignment
1
Impact
_
xb = 40 m/s
mb = 30 kg
m
k
A body of mass mb = 30 kg hits an undamped SDOF system, of unknown
_
characteristics k and m, with velocity xb = 40 m s1 .
After collision the two masses are glued
Dynamics of
Structures
Giacomo Bo
An Introduction to Dynamics of Structures
Giacomo Bo
Dipartimento di Ingegneria Strutturale, Politecnico di Milano
March 18, 2012
Outline
Dynamics of
Structures
Giacomo Bo
Dynamics of
Structures
Giacomo Bo
Introduction
Pa
Continuous
Systems, Innite
Degrees of
Freedom
Outline
Giacomo Bo
Continuous Systems, Innite Degrees of
Freedom
Continous
Systems
Continuous
Systems, Innite
Degrees of
Freedom
Giacomo Bo
Continous Systems
Beams in Flexure
Continous
Systems
Beams in Flexure