1)I Introduction to Structural Dynamics SDOF_short Version

7/25/2019 1)I Introduction to Structural Dynamics SDOF_short Version

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INTRODUCTION TO

STRUCTURAL DYNAMICS

Part I SDOF systems

Luca Pel

Universidad Politcnica de Catalunya

[email protected]

ETS CAMINOS, CANALES Y PUERTOS DE BARCELONA

Diseo y Evaluacin Ssmica de Estructuras
mailto:[email protected]:[email protected]


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ETCECCPB-UPC Introduction to Structural Dynamics

CONTENTS

1. INTRODUCTION

EQUATION OF MOTION

2. FREE VIBRATION OF SDOF SYSTEMS

SYSTEM WITH/WITHOUT DAMPING

3. RESPONSE OF SDOF SYSTEMS TO HARMONIC EXCITATIONS

SYSTEM WITH/WITHOUT DAMPING

4. RESPONSE OF SDOF SYSTEMS TO ARBITRARY EXCITATIONS

DUHAMELS INTEGRAL

5. RESPONSE OF SDOF SYSTEMS TO EARTHQUAKES

RESPONSE SPECTRUM

ELASTIC AND DESIGN RESPONSE SPECTRA (EC8)

BEHAVIOUR FACTOR


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1. INTRODUCTION

DEGREE OF FREEDOM

- The representation of the displacements of a given system with distributed

mass in terms of a finite number of displacements allows to greatly simply

the dynamic problem because inertial forces would develop only at these

points.

- The number of displacement components that must be taken into account torepresent the effects of all significant inertial forces of a system is known as the

number of degrees of freedomof the system (DOF).


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MASS-SPRING-DAMPER SYSTEM

- The system can be seen as the combination of three components: the

stiffnesscomponent, the dampingcomponent and the masscomponent.

( )s d I

p t f f f= + +


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EQUATION OF MOTION

Representation of a SDOF system (mass-spring-damper system):


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2. FREE VIBRATION OF SDOF SYSTEMS

Free vibration:vibration (caused by a disturbance) without any external dynamic excitation

(p(t)=0).


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Natural period of vibration, natural frequency and natural circular frequency:


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SYSTEM WITHOUT DAMPING

Analytical solution of the differential equation:


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SYSTEM WITH DAMPING

The solution of the differential equation can be expressed as

2cr n

c c

c m

= =damping

ratio

>1 overdamped systems (no oscillation)

=1 critically damped systems (no oscillation)


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The damping ratio of structures of interest (buildings, bridges, dams, etc.)

is less than 0.10 (or 10%) underdamped systems

Frequency of damped vibration D(damping lowers the frequency):

>1 overdamped systems (no oscillation)

=1 critically damped systems (no oscillation)


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Typical values of damping ratio in civil engineering structures:

Therefore, in civil engineering structures normally

Steel structures with welded joints or bolted

joints with friction connections

Prestressed reinforced concrete

Uncracked reinforced concrete

2 % - 3%

Cracked reinforced concrete 3% - 5%

Steel structures with bolted joints with

bearing connections

Wood structures

5% - 7%

D


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The equation of motion is now given by:


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SYSTEM WITHOUT DAMPING

The term gives

an oscillation at theexcit ing frequency:

forced orsteady-

state vibration.

The and

terms give an

oscillation at the

systems frequency

transient

vibration.

sin t

sin nt cos nt

r = 0.2


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Resonant frequency is the forcing frequency at which the dynamic

amplification of the displacement is maximum.

For an undamped system (=0) is the natural frequency n.

Video: SDOF Resonance Vibration Test

http://www.youtube.com/watch?v=LV_UuzEznHs

Video: Tacoma Bridge collapse due to resonance:https://www.youtube.com/watch?v=3mclp9QmCGs
http://www.youtube.com/watch?v=LV_UuzEznHshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttps://www.youtube.com/watch?v=3mclp9QmCGshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHshttp://www.youtube.com/watch?v=LV_UuzEznHs


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SYSTEM WITH DAMPING

The transient

response

decays

exponentiallywith time.

After some time

the steady-state

response

dominates.


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4. RESPONSE OF SDOF SYSTEMS TO ARBITRARY EXCITATIONS

In the differential equation of motion,p(t) is an arbitrary deterministic excitation:

p(t) can be interpreted as a sequence of impulses of infinitesimal duration and

the response of the system to p(t) will be the sum of the responses to individualimpulses (this hypothesis limits this procedure to linear elastic systems).


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The response of the system at time t is the sum of the responses to all impulses

up to that time.

convolution integral

magnitude

unit

impulse

response


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A unit impulse causes free vibration to SDOF systems due to the initial

displacement and velocity. The response is given by:


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Considering the unit impulse response function, the Duhamels integral is

used to sum the responses to all impulses.

The Duhamels integral is valid only for linear systems because it is based on

the principle of superposition.

The evaluation of the integral can be based on analytical (if p is a simple

function) or numerical methods (if pis a complicated function).


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TIME-STEPPING METHODS

The analytical solution is usually not possible if the excitation varies arbitrarilywith time or if the system is non-linear. The above problems can be solved by

numerical time-stepping methods for integration of differential equations.

The response is determined at the discrete time instants ti(time i). All values are

assumed to be known at time i. The numerical procedures allow to determine

the response quantities at the time ti+1satisfying the equation

The numerical procedures to be implemented can be divided into 2 main

categories: explicit methods (e.g. Central difference method) and implicit

methods (e.g. Newmarks method).


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NEWMARKS METHOD

The Newmarks method assumes a variation of the acceleration over the timestep [ ti ; ti+1 ].

This variation is controlled by the parameters and which also control the

stability and accuracy of the method. Typically:

The equations of the Newmarks method read


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4. RESPONSE OF SDOF SYSTEMS TO EARTHQUAKES

The equation of motion of a SDOF system subjected to an earthquake reads

dividing by m and substituting c and k

ETCECCPB UPC I t d ti t St t l D i


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RESPONSE SPECTRUM

The peak response of all possible linear SDOF systems to a given earthquake

can be represented in a response spectrum.



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RESPONSE SPECTRUM

The plot of the spectral displacement Sd

against the period Tn

provides the

displacement response spectrum.

The plot of the spectral acceleration Saagainst the period Tnprovides the

acceleration response spectrum .



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The maximum internal (elastic) force can be computed by means of Sd

By expressing k in terms of the mass m, one obtains

It is also possible to define a pseudo-spectral velocity in the form

There is a direct relationship among pseudo-spectral acceleration, pseudo-

spectral velocity and real spectral displacement:

In practice, the pseudo-spectral velocity and pseudo-spectral acceleration are

used instead of the real spectral velocity an spectral acceleration, and no

reference is made to the pseudo characteristic.

ETCECCPB UPC Introduction to Structural Dynamics


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Derivation of Response Spectra from real recordings:

displacement

response spectrum

velocity

response spectrum

acceleration

response spectrum



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The construction of the response spectrum involves the following steps:

1. Definition of the ground acceleration

2. Selection of the parameters Tn and of the SDOF system

3. Computation of the response u(t) of the system due to the ground motion

(e.g. using the Duhamel integral or a numerical method)

4. Determination of the peak value of the response Sd

5. Computation of Svand Saas a function of Sd

6. Repetition of steps 2 to 5 for different values of Tn and

7. Plotting of the response spectrum



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ELASTIC RESPONSE SPECTRUM

The elastic response spectrum it is intended for the design of new structures or

for the seismic safety evaluation of existing structures, to resist futureearthquakes. The definition of the elastic design spectrum parameters should

be based on a probabilistic seismic hazard analysis.

EC8

spectrum



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Equations for each spectral branch of the Eurocode 8 elastic response

spectrum:



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Parameters of the Eurocode 8 elastic response spectrum:

Se(T) : elastic response spectrum

T : vibration period of a linear single-degree-of-freedom system

ag: design ground acceleration on type A ground

TB, TCe TD: limits of the spectral branches

S : soil factor

: damping correction factor



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Soil types:



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Type 1 elastic response spectrum (far-field earthquakes; Ms> 5.5)



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Type 2 elastic response spectrum (near-field earthquakes; Ms< 5.5)



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The short-period portion of the spectrum is dominated by the near-field

earthquake, the long-period portion of the spectrum is dominated by the far-field

earthquake.



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C CC U C t oduct o to St uctu a y a cs

REFERENCES

Chopra A.K. Dynamics of Structures. Prentice Hall, 1995.

Petrini L.; Pinho R.; Calvi G. M. Criteri di progettazione antisismica degli edifici

(in Italian). IUSS Press, 2004.

EN 1998-1. Eurocode 8: Design of structures for earthquake resistance - Part 1:

General rules, seismic actions and rules for buildings.

1)I Introduction to Structural Dynamics SDOF_short Version

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