2014 Ucsm Structural Dynamics 01

8/12/2019 2014 Ucsm Structural Dynamics 01

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ANALISIS DINAMICO DE

ESTRUCTURAS

I ng. Jorge Rosas Espinoza

I ngeniero Civil - Un iversidad Nacional de San Agustin ArequipaPeru

MSc. Student Earthquake Engineeri ng - University at Buff alo, The State University of New York - USA

ANAL ISIS ESTATICO

No aplicable a edificios altos e irregulares.

Masa vertical no uniforme.

Irregularidades torsionales.

Discontinuidad en la rigidez vertical de los elementos.

Irregularidad en la geometria vertical.

Respuesta del edificio en su 1er modo de vibracion (Modo

fundamental).

Distribucion de fuerzas triangular.

Fuerzas de diseno elevadas.


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Esquematiza el comportamiento estructural de una

manera mas exacta.

Edificios altos e irregulares.

Requerido por los codigos de diseno.

Analisis racional y prudente.

Incluye los efectos de los modos de vibracion altos.

Requerido para el estudio del comportamiento no lineal

de estructuras.

ANALI SIS DINAMICO

DESPLAZAM IENTO, VELOCIDAD YACELERACION


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SISTEMAS CON UN GRADO DE L IBERTAD

(Single Degree of F reedom Systems)

Classic SDF System

Mass - SpringDamper System

Single Degree of Freedom System

Mass - Spring

Damper System

RELACION FUERZA DESPLAZAM IENTO(ForceDisplacement Relationships)

L inearl y El astic Systems I nelastic SystemsForceDisplacement relation is linear for small deformations but wouldbecome nonlinear at larger deformations.


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FUERZA DE AMORTIGUAM IENTO

(Damping Force)

In damping, the energy of the vibrating system is dissipated by various mechanisms, andoften more than one mechanism may be present at the same time.In real structures:

Friction at steel connections.

Opening and closing of microcracks in concrete.

Friction between the structure itself and nonstructural elements.

L inear Viscous Damper or Dashpot

It seems impossible to identify or describe mathematically each ofthese energy-dissipating mechanisms in an actual building.

ECUACIONES DE MOVIM IENTO(Equations of Motion)

1)Newtons Second Law of Motion.

Diagrama de cuerpo li bre

(Free Body Diagram)

Ecuacion D iferencial Ordinaria

(Ordinary Dif ferential Equation ODE)


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ECUACIONES DE MOVIM IENTO

(Equations of Motion)

2)DAlembertsPrinciple of Dynamic Equilibrium.


(Free Body Diagram)

Ecuacion D iferencial Ordinaria

(Ordinary Dif ferential Equation ODE)

Natural Frequency Natural Period

VIBRACION LI BRE NO AMORTIGUADA

(Undamped Free - Vibration)

Solution of the form:


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Solution of the Homogeneous

Differential Equation

Initial Conditions

VIBRACION L IBRE NO AMORTIGUADA


VIBRACION LI BRE NO AMORTIGUADA


Vibracion Libre de un Sistema sin Amortiguamiento

(Free Vibration of a System without Damping)


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VIBRACION LI BRE AMORTIGUADA

(Damped Free - Vibration)

Damping Ratio

Solution of the form:

Critical Damping Coefficient

Natural Damped

Frequency




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Solution of the Ordinary

Differential Equation

Initial Conditions

TIPOS DE MOVIM IENTO

(Types of Motion)

Vibracion Libre de sistemas subamortiguados, con amoritguamiento critico y sobreamortiguados

(Free Vibration of underdamped, critically damped and overdamped systems)


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SISTEMAS SUBAMORTIGUADOS

(Underdamped Systems)

Efectos del Amortiguamiento en

vibracion libre

Vibracion libre de Sistemas con Diferentes

niveles de Amortiguamiento

VIBRACION HARMONICA

(Harmonic Vibration)

Harmonic Force

?

Solucion Complementaria u Homogenea Solucion Particular


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Arbitrary Force

?

Time History Response

VIBRACION FORZADA

(Forced Vibration)


(Free Body Diagram)


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I ntegral de Duhamel

Metodos Numericos

: Aceleracion del terr eno (Ground Acceleration)Ground acceleration during

earthquakes varies irregularly to

such an extent that analytical

solution of the equation ofmotion must be rouled out

Deformation Response u(t) of a SDF

System depends only on natural vibrationperiod of the system and its damping ratio.

Once u(t) has been evaluated, the internal

forces can be determined based on theconcept of the Equivalent Static force fs.


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Deformati on Response

Spectrum

Duhamels Integral

(Numeri cal M ethods)

Pseudo-Acceleration:

V is related to the peak value of

the strain energy Eso stored in

the system during the earthquakeby the equation:


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Pseudo-Acceleration

Response Spectrum

Deformati on Response

Spectrum (D)

Good approximationThe difference between the two spectra is small for short-period systems and is of some

significance only for long-period systems with large values of damping.

mA is equal to the peak value of the elastic-resisting force, in contrast is equal to thepeak value of the sum of elastic and damping forces.


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From Response Spectrum (=0.02) for Tn=1.59 sec

The lateral force is:

Properties: I = 7.23 in4, E = 29000 ksi, Assume that

=2 %

From Response Spectrum (=0.02) for Tn=1.59 sec

The shear force is:

Properties: E = 3000 ksi, Each column and beam

has a 10-in-square cross section.

Assume that =2 %

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2014 Ucsm Structural Dynamics 01