03 Ce225 Lecture Overview of Structural Dynamics

8/11/2019 03 Ce225 Lecture Overview of Structural Dynamics

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Structural Dynamics

A subject of study for the analysis and

design of structures subjected to

excitations (forces or motions) which are

time-dependent.

Prerequisites:

Static theory of structures

Matrix methods

Computer coding techniques


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Structural Dynamics

Emphasis

Hand-solution procedure

Computer solution


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Bridge Vibration Due to Traffic


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EarthquakeLoading of

a free

standing

water tank


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Mode 1Vibration


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Structural Project

Structure

Geometry

Loads

Material

Properties

Structural

Analysis

Structural

Design

Construction


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Loads

Static Loading

Dynamic Loading


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Static Analysis

Static Analysis is

determining forces

determining displacements

at any location

on a structure subjected to static loads.


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Dynamic Analysis

Dynamic Analysis is

determining forces

determining displacements

at any location

on a structure subjected to dynamic loads.


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OVERVIEW OF STRUCTURAL DYNAMICS

Structural Dynamicssubject of study for theanalysis and design of structures subjectedto excitations (force or motions) which are

time-dependent.

Dynamic connotes t ime-varying.

Dynamic load means any load in which itsmagnitude, direction, and/or position varieswith time.


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Basic Difference Between Static Loading and

Dynamic Loading


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Dynamic Loading

Deflection and internal forces can

be obtained using principles of

force equilibrium


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Dynamic Loading

Deflection depends upon p(t) and

internal forces

Internal forces must equilibrate

not only p(t) but also inertialforces.


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Thus: structural response (resulting stresses

and deflections) is also time-varying or

dynamic.


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Complications in dynamic behavior:

-inertia forces give structural displacements

and structural displacements give inertia

forces.

there is a closed cycle of cause and

effect; can be dealt with directly only by

formulating the problem in terms of

differential equations.


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Complications:

-mass of the beam is distributed

continuously along its length, the

displacements and accelerations must be

defined for each point along the axis ifinertia forces are to be completely defined

- in this case, analysis must be formulated in

terms of partial differential equationsbecause the position along the span and

the time must be taken as independent

variables.


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Simplifications:

- If mass of the beam were concentrated in

a series of discrete points or lumps, the

analysis problem would be greatlysimplified because inertial forces could be

developed only at these mass points. It is

necessary to define the displacements andaccelerations only at these discrete points.


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Objective of Structural Dynamic Analysis:

- to present methods for analyzing thestresses and deflections of structures when

subjected to dynamic loading.


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In analysis of linear structure, static loadings

are distinguished; response of each type is

evaluated separately; response

components are superposed to obtain thetotal effect.

TOTAL = STATIC + DYNAMIC


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Static loading condition may be looked upon

merely as a special form of dynamic

loading


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Two approaches for evaluating structural

response to dynamic loads:

a) Deterministic approach

b) Nondeterministic approach


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Deterministic Analysis

the time variation of loading is fully

known, even though it is fully oscillatory

or irregular in character (prescribeddynamic loading)

Example: periodic loading

nonperiodic loading


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periodic loading


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nonperiodic loading


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Nondeterministic Analysis

the time variation of loading is not

completely known but can be defined in

a statistical sense (random dynamicloading)

example: wind or earthquake loading


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In general, structural response to any

dynamic loading is expressed basically in

terms of displacements of the structure.

Stresses, strains, internal forces, etc. are

usually obtained as a secondary result.


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Essential characteristics of a dynamic

problem:

1. time-varying nature

2. Inertial forces resist the accelerations of

the structure

Structural

displacements

Inertia

forces


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Degree of Freedom (DOF)

is the number of displacement

components which must be considered in

order to represent the effects of allsignificant inertia forces on the structure.

- the number of independent coordinates

necessary to describe the motion of thesystem.


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Modeling

- The art or process of writing down an equation or systemof equations to describe the motion of a physical device

Model

- The tool that facilitates the mathematical formulation of

the geometry and behavior characteristics of the structureModel and Analysis Tools

- Intended to quantify structural response (displacements,forces, deformations, etc.)

Discretization- The formulation of a mathematical model to describe the

geometric domain of a prototype structure.


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MODELING

Real Structure Idealization FurtherIdealization

k

m

massless


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Lumped-mass idealization

If 3 mass points could move only in thevertical direction (u1, u2, u3:DOF = 3)


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If, in addition, finite rotational inertia areconsidered in each mass point (u1, u2, u3,

1, 2, 3: DOF = 6)


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If, in addition, axial distortion is significant ineach mass point (DOF = 9)


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If structure could deform in three-dimensional space, each mass would have

6 DOF (DOF = 18)


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If mass in continuously distributed, thenumber of DOF is infinite.


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Lumped-mass idealization- Simple means of limiting the degrees of

freedom

- most effective in structures in which a large

proportion of the total mass is actually

concentrated in a few discrete points.


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Full 3D Frame Model

1

35

29

70105

420

DOF = 420 x 6

= 2,52081522

12 floors x 6 bays x 4 bays

=A x B2,520

2,520

No. of Coefficients = 2,520 x 2,520= 6,354,000

No. of Bytes = 8 x 6.35 MBytes

= 50 MBytes


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RIGID FLOOR DIAPHRAGM (RFD)

ASSUMPTION

DOF = 12 x 3= 36

12 floors x 3 DOFs / Floor

=A x B36

36

No. of Bytes (RFD) = 36 x 36 x8

= 10,368 Bytes

No. of Bytes (full 3D) = 50,000,000 Bytes

= 5,000 times more

1

23


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Idealized inelastic column response models


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Various levels of discretization are possible within the

mathematical model development ranging from lumped parameter

model to structural component models to detailed finite element

models.


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OVERVIEW OF

STRUCTURAL

DYNAMICS


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03 Ce225 Lecture Overview of Structural Dynamics

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