En Curs01 Dsis

7/29/2019 En Curs01 Dsis

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Structural Dynamics and EarthquakeEngineering

Course 1

Introduction.Single degree of freedom systems: Equations ofmotion, problem statement, solution methods.

Course notes are available for download athttp://cemsig.ct.upt.ro/astratan/didactic/dsis/

Dynamics of structures

Dynamics of

structures determination ofresponse ofstructures underthe effect of

dynamic loading Dynamic load is

one whosemagnitude,direction, senseand point ofapplication changesin time

u(t)

p(t)

equipment

with

rotating

mass

u(t)

p(t)

propeller of

a ship

u(t)

p(t)

pressure on

a building

due to blast

u(t)

p(t)

earthquake


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Single degree of freedom systems

Simple structures: mass m

stiffness k

Objective: find out response of SDOF system under theeffect of:

a dynamic load acting on the mass

a seismic motion of the base of the structure

The number of degree of

freedom (DOF) necessaryfor dynamic analysis of a

structure is the number ofindependent displacementsnecessary to define thedisplaced position of

masses with respect totheir initial position

k

m

Single degree of freedomsystems (SDOF)

Single degree of freedom systems

One-storey frame =

mass component

stiffness component

damping component

Number of dynamic degrees of freedom = 1

Number of static degrees of freedom = ?


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Force-displacement relationship

Force-displacement relationship

Linear elastic system:

elastic material

first order analysis

Inelastic system: plastic material

First-order or second-order analysis

Sf k u=

( ),S Sf f u u=


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Damping force

Damping: decreasing with time of amplitude of vibrationsof a system let to oscillate freely

Cause: thermal effect of elastic cyclic deformations of thematerial and internal friction

Damping

Damping in real structures:

friction in steel connections

opening and closing of microcracks in r.c. elements

friction between structural and non-structural elements

Mathematical description of these componentsimpossible

Modelling of damping in real structures equivalent viscous damping


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Damping

Relationship between damping forceand velocity:

c- viscous damping coefficientunits: (Force x Time / Length)

Determination of viscous damping:

free vibration tests

forced vibration tests

Equivalent viscous damping modelling of the energydissipated by the structure in the elastic range

Df c u=

Equation of motion for an external force

Newtons second law of motion

D'Alambert principle

Stiffness, damping and mass components


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Equation of motion: Newtons 2nd law of motion

Forces acting on mass m: external force p(t)

elastic (or inelastic) resisting force fS damping force fD

External force p(t), displacement u(t), velocity andacceleration are positive in the positive direction ofthe xaxis

Newtons second law of motion:

( )u t( )u t

S Dp f f mu =

S Dmu f f p+ + =

( )mu cu ku p t + + =

Equation of motion: D'Alambert principle

Inertial force equal to the product between force and acceleration

acts in a direction opposite to acceleration

D'Alambert principle: a system is in equilibrium at eachtime instant if al forces acting on it (including the inertiaforce) are in equilibrium

I S Df f f p+ + =

If mu=

S Dmu f f p+ + =

( )mu cu ku p t + + =


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Equation of motion:stiffness, damping and mass components

Under the external force p(t), the system state isdescribed by

displacement u(t)

velocity

acceleration

System = combination of three pure components:

stiffness component

damping component

mass component

External force p(t) distributed to the three components

( )u t

( )u t

If mu= Df c u=

Sf k u=

I S Df f f p+ + =

SDOF systems: classical representation


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Equation of motion: seismic excitation

Dynamics of structures in the case of seismic motion

determination of structural response under the effect ofseismic motion applied at the base of the structure

Ground displacement ug Total (or absolute) displacement of the mass ut

Relative displacement between mass and ground u

( ) ( ) ( )t

gu t u t u t = +


D'Alambert principle of dynamic equilibrium

Elastic forces relative displacement u

Damping forces relative displacement u

Inertia force total displacement ut

0I S Df f f+ + =

Df c u=

Sf k u=

tIf mu=

0t

mu cu ku+ + =

( ) ( ) ( )tg

u t u t u t = +

gmu cu ku mu+ + =


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Equation of motion in the case of an external force

Equation of motion in the case of seismic excitation

Equation of motion for a system subjected to seismicmotion described by ground acceleration is identicalto that of a system subjected to an external force

Effective seismic force

gmu cu ku mu+ + =

( )mu cu ku p t + + =

gmu gu

( ) ( )eff gp t mu t=

Problem formulation

Fundamental problem in dynamics of structures:determination of the response of a (SDOF) system undera dynamic excitation

a external force

ground acceleration applied to the base of the structure

"Response" any quantity that characterizes behaviourof the structure

displacement

velocity

mass acceleration

forces and stresses in structural members


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Determination of element forces

Solution of the equation of motion of the SDOF system

displacement time history

Displacements forces in structural elements Imposed displacements forces in structural elements

Equivalent static force: an external static force fSthat producesdisplacements udetermined from dynamic analysis

Forces in structural elements by static analysis of the structuresubjected to equivalent seismic forces fS

( )u t

( ) ( )s

f t ku t=

Combination of static and dynamic response

Linear elastic systems:

superposition of effects possible total response can be determinedthrough the superposition of theresults obtained from:

static analysis of the structure under

permanent and live loads, temperatureeffects, etc.

dynamic response of the structure

Inelastic systems: superposition of

effects NOT possible dynamicresponse must take account ofdeformations and forces existing inthe structure before application ofdynamic excitation


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Solution of the equation of motion

Equation of motion of a SDOF system

differential linear non-homogeneous equation of secondorder

In order to completely define the problem:

initial displacement

initial velocity

Solution methods: Classical solution

Duhamel integral

Numerical techniques

( ) ( ) ( ) ( )mu t cu t ku t p t + + =

(0)u

(0)u

Classical solution

Complete solution u(t)of a linear non-homogeneousdifferential equation of second order is composed of

complementary solution uc(t)and

particular solution up(t)u(t)= uc(t)+up(t)

Second order equation 2 integration constants initialconditions

Classical solution useful in the case of

free vibrations

forces vibrations, when dynamic excitation is defined analytically


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Classical solution: example

Equation of motion of an undamped (c=0) SDOF systemexcited by a step force p(t)=p0, t0:

Particular solution:

Complementary solution:

where A and Bare integration constants and

The complete solution

Initial conditions: for t=0 we have and

the eq. of motion

0mu ku p+ =0( )p

pu t

k=

( ) cos sinc n nu t A t B t = +

nk m =

0( ) cos sinn n

pu t A t B t

k = + +

(0) 0u = (0) 0u =

0 0p

A Bk

= =0( ) (1 cos )n

pu t t

k=

Duhamel integral

Basis: representation of the dynamic excitation as asequence of infinitesimal impulses

Response of a system excited by the force p(t)at time tsum of response of all impulses up to that time

Applicable only to "at rest" initial conditions

Useful when the force p(t)

is defined analytically

is simple enough to evaluate analytically the integral

0

1

( ) ( )sin[ ( )]

t

nnu t p t d m

=

(0) 0u = (0) 0u =


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Duhamel integral: example

Equation of motion of an undamped (c=0) SDOF system,excited by a ramp force p(t)=p0, t0:

Equation of motion

0mu ku p+ =

0( ) (1 cos )np

u t tk

=

0 (1 cos )np

tk

=

00

0 0

cos ( )1( ) sin[ ( )]

tt

n

n

n n n

p tu t p t d

m m

=

=

= = =

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