P-752 Unit 2 Fundamentals

Course Introduction
Slide set developed by Finley A. Charney, Ph.D., P.E.
Instructional Material Complementing FEMA P-751, Design Examples
Fundamentals - 1
Overview
Fundamental Concepts
Ground Motions and Their Effects
Structural Dynamics of Linear SDOF Systems
Response Spectra
Structural Dynamics of Simple MDOF Systems
Inelastic Behavior
Structural Design
Fundamentals - 2
Fundamental Concepts (1)
Ordinarily, a large earthquake produces the most severe loading that a

building is expected to survive. The probability that failure will occur is
very real and is greater than for other loading phenomena. Also, in the
case of earthquakes, the definition of failure is altered to permit certain
types of behavior and damage that are considered unacceptable in
relation to the effects of other phenomena.
The levels of uncertainty are much greater than those encountered in

the design of structures to resist other phenomena. The high uncertainty
applies both to knowledge of the loading function and to the resistance
properties of the materials, members, and systems.
The details of construction are very important because flaws of no

apparent consequence often will cause systematic and unacceptable
damage simply because the earthquake loading is so severe and an
extended range of behavior is permitted.
Fundamentals - 3
Fundamental Concepts (2)
During an earthquake the ground shakes violently in all directions.

Buildings respond to the shaking by vibration, and the movements
caused by the vibration and the ground motion induce inertial forces
throughout the structure.
In most parts of the country the inertial forces are so large that it is not
economical to design a building to resist the forces elastically. Thus
inelastic behavior is necessary, and structures must be detailed to
survive several cycles of inelastic behavior during an earthquake.
The structural analysis that is required to exactly account for the

dynamic loading and the inelastic response is quite complex, and is too
cumbersome for most projects. The NEHRP Provisions and ASCE 7
provide simplified approximate analysis approaches that overcome
these difficulties.
Rules for detailing structures for seismic resistance are provided by

standards such as ACI 318 and the AISC Specification and the AISC
Seismic Provisions
Fundamentals - 4
Overview
Response Spectra
Inelastic Behavior
Structural Design
Fundamentals - 5
Seismic Activity on Earth
Fundamentals - 6
Tectonic Plates
Fundamentals - 7
Section of Earth Crust at Ocean Rift Valley
Fundamentals - 8
Section of Earth Crust at Plate

Boundary (Subduction Zone)
Fundamentals - 9
Fault Features
Fault trace
Strike angle
Fault
Dip angle
Fundamentals - 10
Faults and Fault Rupture
Epicenter
Rupture surface
Hypocenter
(focus)
Fault plane
Fundamentals - 11
Types of Faults
Strike Slip
(Left Lateral)
Normal
Strike Slip
(Right Lateral)
Reverse (Thrust)
Fundamentals - 12
Seismic Wave Forms (Body Waves)
Compression Wave
(P Wave)
Shear Wave
(S Wave)
Fundamentals - 13
Seismic Wave Forms (Surface Waves)
Love Wave
Rayleigh Wave
Fundamentals - 14
Arrival of Seismic Waves
P Waves
S Waves
Love Waves
Fundamentals - 15
Effects of Earthquakes
Ground Failure
Rupture
Landslide
Liquefaction
Lateral Spreading
Tsunami
Seiche
Ground Shaking
Fundamentals - 16
Recorded Ground Motions
Fundamentals - 17
Shaking at the Holiday Inn During the

1971 San Fernando Valley EQ
Fundamentals - 18
Overview
Response Spectra
Inelastic Behavior
Structural Design
Fundamentals - 19
NEHRP Seismic Hazard Maps

T=0.2 Seconds
Probabilistic / Deterministic (Separate Maps)

Uniform Risk (Separate Maps)
Spectral Contours (PGA, 0.1, 0.2 sec)
5 % Damping
Site Class B/C Boundary
Maximum Direction Values
T=1.0 Seconds
Fundamentals - 20
Structural Dynamics of SDOF Systems

f I (t )
0.5 f S ( t )
f D (t )
F (t )
0.5 f S ( t )
f I (t ) + f D (t ) + fS (t ) = F (t )
( t ) + c u ( t ) + k u( t ) = F ( t )
mu
Fundamentals - 21
Mass
Internal Force
Mass
M
1.0
Acceleration
Includes all dead weight of structure

May include some live load
Has units of force/acceleration
Fundamentals - 22
Damping
Damping Force
Linear Viscous Damping
C
1.0
Velocity
In absence of dampers, is called inherent damping

Usually represented by linear viscous dashpot
Has units of force/velocity
Fundamentals - 23
Damping
Damping Force
Damping and Energy Dissipation
AREA =
ENERGY
DISSIPATED
Displacement
Damping vs displacement response is

elliptical for linear viscous damper.
Fundamentals - 24
Spring Force
Stiffness
Elastic Stiffness
K
1.0
Displacement
Includes all structural members

May include some seismically nonstructural members
Requires careful mathematical modelling
Has units of force/displacement
Fundamentals - 25
Spring Force
Stiffness
Inelastic Behavior
AREA =
ENERGY
DISSIPATED
Displacement
Is almost always nonlinear in real seismic response

Nonlinearity is implicitly handled by codes
Explicit modelling of nonlinear effects is possible
(but very difficult)
Fundamentals - 26
Undamped Free Vibration

Equation of motion:
Initial conditions:
Assume:
u( t ) = A sin(t ) + B cos(t )
A=
u0
u (t ) =
u0
Solution:
( t ) + k u( t ) = 0
mu
u0 u0
B = u0
k
m
sin(t ) + u0 cos(t )
Fundamentals - 27
Undamped Free Vibration (2)

Displacement, inches
u0
T = 0.5 sec
1.0
3
2
1
0
-1
-2
-3
u0
0.0
0.5
1.0
1.5
2.0
Time, seconds
Circular Frequency
(radians/sec)
k
m
Cyclic Frequency
(cycles/sec, Hertz)
f =
2
Period of Vibration
(sec/cycle)
1 2
T=
=
f
Fundamentals - 28
Periods of Vibration of Common Structures

20-story moment resisting frame
T = 2.4 sec
T = 1.3 sec
T = 0.2 sec
20-story steel braced frame

T = 1.6 sec
T = 0.9 sec
T = 0.1 sec
Gravity dam
Suspension bridge
T = 0.2 sec
T = 20 sec
Fundamentals - 29
Damped Free Vibration
( t ) + c u( t ) + k u( t ) = 0
mu
u0
Initial conditions: u0
st
Assume: u( t ) = e
Equation of motion:
Solution:
u( t ) = e
u0 + u0
sin( D t )
u0 cos( D t ) +
D
c
c
=
=
cc
2 m
D = 1
Fundamentals - 30
Damped Free Vibration (2)

c
c
=
=
cc
2 m
cc is the critical damping constant.
is expressed as a ratio (0.0 < < 1.0) in computations.

Sometimes is expressed as a% (0 < < 100%).
Displacement, in
Time, sec
Response of Critically Damped System, =1.0 or 100% critical
Fundamentals - 31
Damped Free Vibration (3)

3
2
1
0
-1
-2
-3
0% Damping
10% Damping
20% Damping
0.0
0.5
1.0
1.5
2.0
Time, seconds
True damping in structures is NOT viscous. However,

for low damping values, viscous damping allows for
linear equations and vastly simplifies the solution.
Fundamentals - 32
Damping in Structures
Welded steel frame
Bolted steel frame
= 0.010
= 0.020
Uncracked prestressed concrete

Uncracked reinforced concrete
Cracked reinforced concrete
= 0.015
= 0.020
= 0.035
Glued plywood shear wall

Nailed plywood shear wall
= 0.100
= 0.150
Damaged steel structure

Damaged concrete structure
= 0.050
= 0.075
Structure with added damping
= 0.250
Fundamentals - 33
Undamped Harmonic Loading and Resonance

(t ) + ku (t ) = p0 sin( t )
mu
80
2 uS
Displacement, in.
40
-40
Linear envelope
-80
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Time, seconds
Fundamentals - 34
Damped Harmonic Loading and Resonance
( t ) + cu ( t ) + k u(t ) = p0 sin( t )
mu
Displacement Amplitude, Inches
50
40
1
Static
2
30
20
10
0
-10
-20
-30
-40
-50
0.00
1.00
2.00
3.00
4.00
5.00
Time, Seconds
Fundamentals - 35
Resonant Response Curve

14.00
Resonance
0.0% Damping
5.0 % Damping
10.0% Damping
25.0 % Damping
Dynamic Response Amplifier
12.00
10.00
8.00
RD =
6.00
1
(1 ) + (2 )
2 2
4.00
2.00
Slowly
loaded
0.00
0.00
0.50
Rapidly
loaded
1.00
1.50
2.00
2.50
3.00
Frequency Ratio,
Fundamentals - 36
General Dynamic Loading
Fourier transform
Duhamel integration
Piecewise exact
Newmark techniques
All techniques are carried out numerically.

Fundamentals - 37
Effective Earthquake Force
Ground Acceleration Response History

GROUND ACC, g
0.40
0.20
0.00
-0.20
-0.40
0.00
1.00
2.00
3.00
4.00
5.00
6.00
TIME, SECONDS
m [ug ( t ) + ur ( t )] + c ur ( t ) + k ur ( t ) = 0
mur ( t ) + c ur ( t ) + k ur ( t ) = mug ( t )
Fundamentals - 38
Simplified SDOF Equation of Motion

mur (t ) + cur (t ) + kur (t ) = mug (t )
Divide through by m:
c
k
ur (t ) + ur (t ) + ur (t ) = ug (t )
m
m
Make substitutions:
c
= 2
m
k
2
=
m
Simplified form:
ur (t ) + 2 ur (t ) + 2ur (t ) = ug (t )
Fundamentals - 39
Use of Simplified Equation of Motion

For a given ground motion, the response history ur(t) is
function of the structures frequency and damping ratio .
Structural frequency
ur (t ) + 2 u r (t ) + u r (t ) = ug (t )
2
Damping ratio
Ground motion acceleration history
Fundamentals - 40
Use of Simplified Equation

Excitation applied to structure
with given and
0.3
0.2
0.1
0
-0.1
SOLVER
-0.2
-0.3
0
10
20
30
40
50
60
Time (sec)
6
Change in ground
motion or structural
parameters and
requires re-calculation
of structural response
Structural Displacement (in)
Ground Acceleration (g's)
0.4
Computed response
4
2
0
-2
-4
Peak displacement
-6
0
10
20
30
40
50
60
Time (sec)
Fundamentals - 41
Creating an Elastic Response Spectrum

An elastic displacement response spectrum is a plot
of the peak computed relative displacement, ur, for an
elastic structure with a constant damping , a varying
fundamental frequency (or period T = 2/ ),
responding to a given ground motion.
5% damped response spectrum for structure
responding to 1940 El Centro ground motion
DISPLACEMENT, inches
16
12
0
0
10
PERIOD, Seconds
Fundamentals - 42
Pseudoacceleration Spectrum
400.0
5% damping
Pseudoacceleration, in/sec
350.0
300.0
250.0
200.0
PSA(T ) 2 D
150.0
100.0
50.0
0.0
0.0
1.0
2.0
3.0
4.0
Period, Seconds
Fundamentals - 43
Pseudoacceleration is Total Acceleration

The pseudoacceleration response spectrum represents the total
acceleration of the system, not the relative acceleration. It is nearly identical
to the true total acceleration response spectrum for lightly damped structures.
400.0
5% damping
Pseudoacceleration, in/sec2
350.0
Peak ground
acceleration
300.0
250.0
200.0
150.0
100.0
50.0
0.0
0.0
1.0
2.0
3.0
4.0
Period, Seconds
Fundamentals - 44
Using Pseudoacceleration to Compute

Seismic Force
400.0
K = 500 k/in
350.0
W = 2,000 k
M = 2000/386.4 = 5.18 k-sec2/in
= (K/M)0.5 =9.82 rad/sec
T = 2/ = 0.64 sec
5% critical damping
Pseudoacceleration, in/sec2
Example Structure
300.0
250.0
200.0
150.0
100.0
50.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Period, Seconds
At T = 0.64 sec, pseudoacceleration = 301 in./sec2

Base shear = M x PSA = 5.18(301) = 1559 kips
Fundamentals - 45
Response Spectra for 1971 San Fernando

Valley EQ (Holiday Inn)
Fundamentals - 46
Averaged Spectrum and Code Spectrum
Fundamentals - 47
NEHRP/ASCE 7 Design Spectrum
Constant Acceleration
Constant Velocity
Constant Displacement
Fundamentals - 48
Overview
Response Spectra
Inelastic Behavior
Structural Design
Fundamentals - 49
MDOF Systems
u1
u2
u3
Fundamentals - 50
Analysis of Linear MDOF Systems

MDOF Systems may either be solved step by step through time
by using the full set of equations in the original coordinate
system, or by transforming to the Modal coordinate system,
analyzing all modes as SDOF systems, and then converting back
to the original system. In such a case the solutions obtained are
mathematically exact, and identical. This analysis is referred to
as either Direct (no transformation) or Modal (with transformation)
Linear Response History Analysis. This procedure is covered in
Chapter 16 of ASCE 7.
Alternately, the system may be transformed to modal

coordinates, and only a subset (first several modes) of equations
be solved step by step through time before transforming back to
the original coordinates. Such a solution is approximate. This
analysis is referred to as Modal Linear Response History
Analysis. This procedure is not directly addressed in ASCE 7
(although in principle, Ch. 16 could be used)
Fundamentals - 51
Analysis of Linear MDOF Systems

Another alternate is to convert to the modal coordinates, and
instead of solving step-by-step, solve a subset (the first several
modes) of SDOF systems system using a response spectrum.
Such a solution is an approximation of an approximation. This
analysis is referred to as Modal Response Spectrum Analysis.
This procedure is described in Chapter 12 of ASCE 7.
Finally, the equivalent lateral force method may be used, which in

essence, is a one-mode (with higher mode correction) Modal
Response Spectrum Analysis. This is an approximation of an
approximation of an approximation (but is generally considered to
be good enough for design.) The Provisions and ASCE 7 do
place some restrictions on the use of this method.
Fundamentals - 52
Overview
Response Spectra
Inelastic Behavior
Structural Design
Fundamentals - 53
Basic Base Shear Equations in

NEHRP and ASCE 7
SDS
Cs =
(R /Ie )
V = CsW
SD1
Cs =
T (R /Ie )
SDS and SD1 are short and one second (T=0.2 s and 1.0 s)Design
Basis Spectral Accelerations, including Site Effects
Ie is the Importance Factor
R is a Response Modification Factor, representing Inelastic
Behavior (Ductility, Over-strength, and a few other minor
ingredients).
Fundamentals - 54
Building Designed for Wind or Seismic Load

Building properties:
Moment resisting frames
Density = 8 pcf
Period T = 1.0 sec
Damping = 5%
Soil Site Class B
62.5
90
120
Total wind force on 120 face = 406 kips

Total wind force on 90 face = 304 kips
Total ELASTIC earthquake force (in each direction) = 2592 kips
Fundamentals - 55
Comparison of EQ vs Wind
VEQ
VW 120
2592
=
= 6 .4
406
VEQ
VW 90
2592
=
= 8.5
304
ELASTIC earthquake forces 6 to 9 times wind!

Virtually impossible to obtain economical design
Fundamentals - 56
How to Deal with Huge EQ Force?
Pay the premium for remaining elastic

Isolate structure from ground (seismic isolation)
Increase damping (passive energy dissipation)
Allow controlled inelastic response
Historically, building codes use inelastic response procedure.
Inelastic response occurs through structural damage (yielding).
We must control the damage for the method to be successful.
Fundamentals - 57
Nonlinear Static Pushover Analysis
Force, kips
Actual
600
Idealized
400
Over-Strength
First Yield
200
1.0
2.0
3.0
4.0
5.0
Fundamentals - 58
Mathematical Model and Ground Motion

600
Idealized SDOF Model
Force, kips
Fy=500 k
K2=11 k/in
400
200
K1=550 k/in
2.0
1.0
4.0
3.0
5.0
Displacement, in.
El Centro Ground Motion

Acceleration, g
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
10
12
14
16
18
20
Time, Seconds
Fundamentals - 59
Results of Nonlinear Analysis

Maximum
displacement:
4.79
Maximum
shear force:
542 k
Number of
yield events:
15
Fundamentals - 60
Response Computed by Nonlin
Yield displacement = 500/550 = 0.91 inch

Maximum Displacement 4.79
=
= 5.26
Ductility Demand
Yield Displacement
0.91
Fundamentals - 61
Interim Conclusion (the Good News)

The frame, designed for a wind force which is 15%
of the ELASTIC earthquake force, can survive the
earthquake if:
It has the capability to undergo numerous

cycles of INELASTIC deformation
It has the capability to deform at least 5 to 6
times the yield deformation
It suffers no appreciable loss of strength
REQUIRES ADEQUATE DETAILING
Fundamentals - 62
Interim Conclusion (The Bad News)

As a result of the large displacements
associated with the inelastic deformations,
the structure will suffer considerable
structural and nonstructural damage.
This damage must be controlled by
adequate detailing and by limiting structural
deformations (drift).
Fundamentals - 63
Development of the Equal

Displacement Concept
Concept used by:
IBC
NEHRP
ASCE-7
In association with force based

design concept. Used to predict
design forces and displacements
ASCE 41
In association with static pushover

analysis. Used to predict displacements
at various performance points.
Fundamentals - 64
The Equal Displacement Concept

The displacement of an inelastic system, with
stiffness K and strength Fy, subjected to a
particular ground motion, is approximately equal
to the displacement of the same system
responding elastically.
(The displacement of a system is independent
of the yield strength of the system.)
Fundamentals - 65
Maximum Displacement (in)
Repeated Analysis for Various Yield

Strengths (and constant stiffness)
Elastic = 5.77
Avg. = 5.0 in.

7
6
5
4
3
2
1
0
500
1000
1500
2000
2500
3000
3500
4000
Yield Strength (K)

Ductility Demand
6
5
Inelastic
4
3
2
Elastic
1
0
0
500
1000
1500
2000
2500
3000
3500
4000
Yield Strength (K)

Fundamentals - 66
Constant Displacement Idealization of

Inelastic Response
IDEALIZED BEHAVIOR
3500
3500
3000
3000
2500
2500
Force, Kips
Force, Kips
ACTUAL BEHAVIOR
2000
1500
1000
2000
1500
1000
500
500
0
0
4
Displacement, Inches
Fundamentals - 67
Equal Displacement Idealization

of Inelastic Response
For design purposes, it may be assumed that
inelastic displacements are equal to the
displacements that would occur during an
elastic response.
The required force levels under inelastic response

are much less than the force levels required for
elastic response.
Fundamentals - 68
Equal Displacement Concept of Inelastic

Design
3500
Elastic
Force, Kips
3000
2500
2000
1500
1000
Inelastic
500
0
0
5.77 6
Design for this force
Design for this displacement
Fundamentals - 69
Force, Kips
Key Ingredient: Ductility

3500
3000
2500
Elastic
2000
1500
1000
500
0
0 dy=0.91 2
Inelastic
4 du=5.77 6
Ductility supply MUST BE > ductility demand =

5.77
= 6.34
0.91
Fundamentals - 70
Application in Principle
3500
FE
Force, Kips
3000
2500
2000
1500
1000
FI
500
0
0
dR
dI 6
Using response spectra, estimate elastic force demand FE

Estimate ductility supply, , and determine inelastic force demand
FI = FE /. Design structure for FI.
Compute reduced displacement. dR, and multiply by to obtain
true inelastic displacement, dI. Check Drift using dI.
Fundamentals - 71
Application in Practice
(NEHRP and ASCE 7)
Use basic elastic spectrum but, for strength,
divide all pseudoacceleration values by R, a
response modification factor that accounts for:
Anticipated ductility supply

Overstrength
Damping (if different than 5% of critical)
Past performance of similar systems
Redundancy
Fundamentals - 72
Ductility/Overstrength
First Significant Yield
First
Significant
Yield
Force
Design
Strength
Displacement
Fundamentals - 73
First Significant Yield

and Design Strength
First Significant Yield is the level of force

that causes complete plastification of at least
the most critical region of the structure (e.g.,
formation of the first plastic hinge).
The design strength of a structure is equal to
the resistance at first significant yield.
Fundamentals - 74
Overstrength
Force
Apparent
Strength
Over Strength
Design
Strength
Displacement
Fundamentals - 75
Sources of Overstrength
Sequential yielding of critical regions

Material overstrength (actual vs specified yield)
Strain hardening
Capacity reduction ( ) factors
Member selection
Structures where the proportioning is controlled
by the seismic drift limits
Fundamentals - 76
Definition of Overstrength Factor

Overstrength Factor =
Apparent Strength
Design Strength
Force
Apparent Strength
Overstrength
Design Strength
Displacement
Fundamentals - 77
Definition of Ductility Reduction Factor Rd
Fundamentals - 78
Definition of Response Modification

Coefficient R
Overstrength Factor =
Ductility Reduction Rd =
Apparent Strength
Design Strength
Elastic Strength Demand
Apparent Strength
Elastic Strength Demand

R=
Design Strength
= Rd
Fundamentals - 79
Definition of Response Modification

Coefficient R
Fundamentals - 80
Definition of Deflection Amplification

Factor Cd
Fundamentals - 81
Example of Design Factors for

Reinforced Concrete Structures
R
Cd
Special Moment Frame
5.5
Intermediate Moment Frame

Ordinary Moment Frame
5
3
3
3
4.5
2.5
Special Reinforced Shear Wall

Ordinary Reinforced Shear Wall
Detailed Plain Concrete Wall
Ordinary Plain Concrete Wall
5
4
2
1.5
2.5
2.5
2.5
2.5
5.0
4.0
2.0
1.5
Fundamentals - 82
Design Spectra as Adjusted

for Inelastic Behavior
1.2
SDS=1.0g
S DS
CS =
R/I
S D1
CS =
T(R / I )
Acceleration, g
1.0
SD1=0.48g
R=1
R=2
0.8
R=3
0.6
R=4
R=6
R=8
0.4
0.2
0.0
0
Period, Seconds
Fundamentals - 83
Using Inelastic Spectrum to Determine

Inelastic Force Demand
1.2
Pseudoacceleration, g
R=1
1.0
R=4
0.8
0.6
V=0.15W
0.4
0.15
0.2
0.0
0.0
1.0
2.0
3.0
4.0
5.0
Period, Seconds
Fundamentals - 84
Using the Inelastic Spectrum and Cd to

Determine the Inelastic Displacement
Demand
INELASTIC = Cd REDUCED ELASTIC
25
INELASTIC = 3.65 in
Displacement,inches.
R=1
R=4
20
(R=4)*Cd
15
Cd=3.5
10
5
0
0
Period, Seconds
Fundamentals - 85
Overview
Response Spectra
Inelastic Behavior
Structural Design
Fundamentals - 86
Design and Detailing Requirements
Fundamentals - 87
Questions
Fundamentals - 88

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Course Introduction

Slide set developed by Finley A. Charney, Ph.D., P.E.

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

Fundamental Concepts (1)

Ordinarily, a large earthquake produces the most severe loading that a

The levels of uncertainty are much greater than those encountered in

The details of construction are very important because flaws of no

Instructional Material Complementing FEMA P-751, Design Examples

Fundamental Concepts (2)

During an earthquake the ground shakes violently in all directions.

The structural analysis that is required to exactly account for the

Rules for detailing structures for seismic resistance are provided by

Instructional Material Complementing FEMA P-751, Design Examples

Seismic Activity on Earth

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

Section of Earth Crust at Ocean Rift Valley

Instructional Material Complementing FEMA P-751, Design Examples

Section of Earth Crust at Plate

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

Faults and Fault Rupture

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

Seismic Wave Forms (Body Waves)

Instructional Material Complementing FEMA P-751, Design Examples

Seismic Wave Forms (Surface Waves)

Instructional Material Complementing FEMA P-751, Design Examples

Arrival of Seismic Waves

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

Recorded Ground Motions

Instructional Material Complementing FEMA P-751, Design Examples

Shaking at the Holiday Inn During the

Instructional Material Complementing FEMA P-751, Design Examples

Instructional Material Complementing FEMA P-751, Design Examples

NEHRP Seismic Hazard Maps

Probabilistic / Deterministic (Separate Maps)

Instructional Material Complementing FEMA P-751, Design Examples

Structural Dynamics of SDOF Systems

Includes all dead weight of structure

Linear Viscous Damping

In absence of dampers, is called inherent damping

Damping and Energy Dissipation

Damping vs displacement response is

Instructional Material Complementing FEMA P-751, Design Examples

Includes all structural members

Is almost always nonlinear in real seismic response

Undamped Free Vibration

Instructional Material Complementing FEMA P-751, Design Examples

Undamped Free Vibration (2)

Instructional Material Complementing FEMA P-751, Design Examples

Periods of Vibration of Common Structures

20-story steel braced frame

Instructional Material Complementing FEMA P-751, Design Examples

Damped Free Vibration

Instructional Material Complementing FEMA P-751, Design Examples

Damped Free Vibration (2)