Post on 03-Jun-2018
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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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OVERVIEW OF STRUCTURAL DYNAMICS
Basic Difference Between Static Loading and
Dynamic Loading
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OVERVIEW OF STRUCTURAL DYNAMICS
Basic Difference Between Static Loading and
Dynamic Loading
Deflection and internal forces can
be obtained using principles of
force equilibrium
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OVERVIEW OF STRUCTURAL DYNAMICS
Basic Difference Between Static Loading and
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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OVERVIEW OF STRUCTURAL DYNAMICS
Thus: structural response (resulting stresses
and deflections) is also time-varying or
dynamic.
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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
Objective of Structural Dynamic Analysis:
- to present methods for analyzing thestresses and deflections of structures when
subjected to dynamic loading.
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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
Static loading condition may be looked upon
merely as a special form of dynamic
loading
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OVERVIEW OF STRUCTURAL DYNAMICS
Two approaches for evaluating structural
response to dynamic loads:
a) Deterministic approach
b) Nondeterministic approach
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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
periodic loading
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OVERVIEW OF STRUCTURAL DYNAMICS
nonperiodic loading
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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
If 3 mass points could move only in thevertical direction (u1, u2, u3:DOF = 3)
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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
If, in addition, finite rotational inertia areconsidered in each mass point (u1, u2, u3,
1, 2, 3: DOF = 6)
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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
If, in addition, axial distortion is significant ineach mass point (DOF = 9)
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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
If structure could deform in three-dimensional space, each mass would have
6 DOF (DOF = 18)
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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
If mass in continuously distributed, thenumber of DOF is infinite.
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OVERVIEW OF STRUCTURAL DYNAMICS
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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OVERVIEW OF STRUCTURAL DYNAMICS
Lumped-mass idealization
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OVERVIEW OF STRUCTURAL DYNAMICS
Idealized inelastic column response models
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OVERVIEW OF STRUCTURAL DYNAMICS
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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