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7/29/2019 En Curs01 Dsis
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1
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 = +
Equation of motion: seismic excitation
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: seismic excitation
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
=
=
= = =