Lesson 5 Structural Dynamics
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Transcript of Lesson 5 Structural Dynamics
U.S. Army Corps of Engineers Structural Dynamics 1
Lesson 5
Structural Dynamics
U.S. Army Corps of Engineers Structural Dynamics 2
Lesson Objectives
Upon conclusion, participants should have:
1. A clear understanding of seismic structural response in terms of structural dynamics
2. An appreciation that code-based seismic design provisions are based on the principles of structural dynamics
U.S. Army Corps of Engineers Structural Dynamics 3
Part I
Linear Single Degree of Freedom Systems
U.S. Army Corps of Engineers Structural Dynamics 4
Structural Dynamics of SDOF Systems: Topic Outline
Equations of Motion for SDOF Systems
Structural Frequency and Period of Vibration
Behavior under Dynamic Load
Dynamic Amplification
Effect of Damping on Behavior
Linear Elastic Response Spectra
U.S. Army Corps of Engineers Structural Dynamics 5
Mass
Stiffness
Damping
F t u t( ), ( )
t
F(t)
t
u(t)
Idealized Single Degree of Freedom System
U.S. Army Corps of Engineers Structural Dynamics 6
F t( )f tI ( )
f tD( )0 5. ( )f tS0 5. ( )f tS
mu t c u t k u t F t( ) ( ) ( ) ( )
)t(F)t(f)t(f)t(f SDI =++
Equation of Dynamic Equilibrium
U.S. Army Corps of Engineers Structural Dynamics 7
MASS
• Includes all dead weight of structure• May include some live load• Has units of FORCE/ACCELERATION
INE
RT
IAL
FO
RC
E
ACCELERATION
1.0
M
Properties of Structural MASS
U.S. Army Corps of Engineers Structural Dynamics 8
DAMPING
• In absence of dampers, is called Natural Damping• Usually represented by linear viscous dashpot• Has units of FORCE/VELOCITY
DA
MP
ING
FO
RC
E
VELOCITY
1.0
C
Properties of Structural DAMPING
U.S. Army Corps of Engineers Structural Dynamics 9
• Includes all structural members• May include some “seismically nonstructural” members• Has units of FORCE/DISPLACEMENT
SP
RIN
G F
OR
CE
DISPLACEMENT
1.0
K
ST
IFF
NE
SS
Properties of Structural STIFFNESS
U.S. Army Corps of Engineers Structural Dynamics 10
• Is almost always nonlinear in real seismic response• Nonlinearity is implicitly handled by codes• Explicit modelling of nonlinear effects is possible
SP
RIN
G F
OR
CE
DISPLACEMENT
ST
IFF
NE
SS
AREA =ENERGYDISSIPATED
Properties of Structural STIFFNESS (2)
U.S. Army Corps of Engineers Structural Dynamics 11
)tcos(u)tsin(u
)t(u ω+ωω
= 00
mu t k u t( ) ( ) 0Equation of Motion:
00uu Initial Conditions:
Solution:
m
k=ω
Undamped Free Vibration
U.S. Army Corps of Engineers Structural Dynamics 12
m
k=ω
π2
ω=f
ω
π2=
1=
fT
Period of Vibration(sec/cycle)
Cyclic Frequency(cycles/sec, Hertz)
Circular Frequency (radians/sec)
-3
-2
-1
0
1
2
3
0.0 0.5 1.0 1.5 2.0
Time, seconds
Dis
pla
ce
me
nt,
in
ch
es
T = 0.5 sec
u0
u01.0
Undamped Free Vibration (2)
U.S. Army Corps of Engineers Structural Dynamics 13
)]tsin(uu
)tcos(u[e)t(u DD
Dt ω
ω
ξω++ω= 00
0ξω
mu t c u t k u t( ) ( ) ( ) 0Equation of Motion:
00uu Initial Conditions:
Solution:
cc
c
m
c=
ω2=ξ 2ξ-1ω=ωD
Damped Free Vibration
U.S. Army Corps of Engineers Structural Dynamics 14
Time, seconds
Displacement, inches
x = Damping ratio
When x = 1.0, the system is called critically damped.
Response of Critically Damped System, = 1.0x or 100% critical
Damping in Structures
U.S. Army Corps of Engineers Structural Dynamics 15
-3
-2
-1
0
1
2
3
0.0 0.5 1.0 1.5 2.0
Time, seconds
Dis
pla
ce
me
nt,
in
ch
es
0% Damping
10% Damping
20% Damping
Damped Free Vibration
U.S. Army Corps of Engineers Structural Dynamics 16
True damping in structures is NOT viscous. However, for low damping values, viscous damping allows for linear equations and vastly simplifies the solution.
Damping in Structures
-4.00
-2.00
0.00
2.00
4.00
-20.00 -10.00 0.00 10.00 20.00
Velocity, In/sec
Da
mp
ing
Fo
rce
, K
ips
Velocity, in/sec.
U.S. Army Corps of Engineers Structural Dynamics 17
Welded Steel Frame x = 0.010Bolted Steel Frame x = 0.020
Uncracked Prestressed Concrete x = 0.015Uncracked Reinforced Concrete x = 0.020Cracked Reinforced Concrete x = 0.035
Glued Plywood Shear wall x = 0.100Nailed Plywood Shear wall x = 0.150
Damaged Steel Structure x = 0.050Damaged Concrete Structure x = 0.075
Structure with Added Damping x = 0.250
Damping in Structures (2)
U.S. Army Corps of Engineers Structural Dynamics 18
Natural Damping
Supplemental Damping
ξ is a structural (material) property,independent of mass and stiffness
critical07to50=ξ %..NATURAL
ξ is a structural property, dependent onmass and stiffness, anddamping constant C of device
critical30to10=ξ %ALSUPPLEMENT
C
Damping in Structures (3)
U.S. Army Corps of Engineers Structural Dynamics 19
)tsin(p)t(uk)t(um ω=+ 0Equation of Motion:
-150
-100
-50
0
50
100
150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, Seconds
Fo
rce
, Kip
s
po=100 kips = 0.25 sec
= Frequency of the Forcing Function
ω
π2=T
ω
T
Undamped Harmonic Loading
U.S. Army Corps of Engineers Structural Dynamics 20
Solution:
)]tsin()t[sin()/(k
p)t(u ω
ω
ω-ω
ωω-1
1= 2
0
)tsin(p)t(uk)t(um ω=+ 0Equation of Motion:
Assume system is initially at rest:
Undamped Harmonic Loading (2)
U.S. Army Corps of Engineers Structural Dynamics 21
Define ω
ω=β
( ))tsin()tsin(k
p)t(u ωβ-ω
β-1
1= 2
0
Static Displacement, The Steady State
Response is always atthe structure’s loading frequency
Transient Response(at structure’s frequency)
LOADING FREQUENCY
Structure’s NATURAL FREQUENCY
Dynamic Magnifier,
Undamped Harmonic Loading
Su
DR
U.S. Army Corps of Engineers Structural Dynamics 22
-10
-5
0
5
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Dis
pla
ce
me
nt,
in.
-10
-5
0
5
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00Dis
pla
ce
me
nt,
in.
-10
-5
0
5
10
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
Dis
pla
ce
me
nt,
in.
-200-100
0
100200
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00F
orc
e, K
ips
rad/secπ2=ωrad/secπ4=ω uS 50. .in50=β .
LOADING,kips
STEADYSTATERESPONSE, in.
TRANSIENTRESPONSE, in.
TOTALRESPONSE, in.
U.S. Army Corps of Engineers Structural Dynamics 23
STEADYSTATERESPONSE, in.
TRANSIENTRESPONSE, in.
TOTALRESPONSE, in.
LOADING, kips
rad/secπ4=ωrad/secπ4≈ω .in 0.5Su990=β .
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in.
-150
-100
-50
0
50
100
150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00F
orc
e, K
ips
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in.
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
Dis
pla
ce
me
nt,
in.
U.S. Army Corps of Engineers Structural Dynamics 24
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
Dis
pla
ce
me
nt,
in.
Suπ2
Linear Envelope
Undamped Resonant Response Curve
U.S. Army Corps of Engineers Structural Dynamics 25
STEADYSTATERESPONSE, in.
TRANSIENTRESPONSE, in.
TOTALRESPONSE, in.
LOADING, kips
rad/secπ4=ωrad/secπ4≈ω uS 50. .in011=β .
-150
-100
-50
0
50
100
150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00F
orc
e, K
ips
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in
.
-500
-250
0
250
500
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in
.
-80
-40
0
40
80
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
Dis
pla
ce
me
nt,
in
.
U.S. Army Corps of Engineers Structural Dynamics 26
STEADYSTATERESPONSE, in.
TRANSIENTRESPONSE, in.
TOTALRESPONSE, in.
LOADING, kips
rad/secπ8=ωrad/secπ4=ω uS 50. .in02=β .
-150
-100
-50
0
50
100
150
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00F
orc
e, K
ips
-6
-3
0
3
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in
.
-6
-3
0
3
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Dis
pla
ce
me
nt,
in
.
-6
-3
0
3
6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time, seconds
Dis
pla
ce
me
nt,
in
.
U.S. Army Corps of Engineers Structural Dynamics 27
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio
Mag
nifi
catio
n F
acto
r 1/
(1-2
)
Resonance
SlowlyLoaded
1.00
RapidlyLoaded
Response Ratio: Steady State to Static(Absolute Values)
U.S. Army Corps of Engineers Structural Dynamics 28
-200
-150
-100
-50
0
50
100
150
200
0.00 1.00 2.00 3.00 4.00 5.00
Time, Seconds
Dis
pla
cem
en
t A
mp
litu
de
, In
che
s
0% Damping %5 Damping
Harmonic Loading at ResonanceEffects of Damping
U.S. Army Corps of Engineers Structural Dynamics 29
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00
Frequency Ratio,
Dy
na
mic
Re
sp
on
se
Am
plif
ier
0.0% Damping
5.0 % Damping
10.0% Damping
25.0 % Damping
222 ξβ2+β-1
1=
)()(RD
Resonance
SlowlyLoaded Rapidly
Loaded
U.S. Army Corps of Engineers Structural Dynamics 30
For system loaded at a frequency less than √2 or 1.414 times its natural frequency, the dynamic response exceeds the static response. This is referred to as Dynamic Amplification.
An undamped system, loaded at resonance, will have an unbounded increase in displacement over time.
Summary Regarding Viscous Dampingin Harmonically Loaded Systems
U.S. Army Corps of Engineers Structural Dynamics 31
Summary Regarding Viscous Dampingin Harmonically Loaded Systems
Damping is an effective means for dissipating energy in the system. Unlike strain energy, which is recoverable, dissipated energy is not recoverable.
A damped system, loaded at resonance, will have a limited displacement over time, with the limit being (1/2x) times the static displacement.
Damping is most effective for systems loaded at or near resonance.
U.S. Army Corps of Engineers Structural Dynamics 32
LOADING YIELDING
UNLOADING UNLOADED
F F
F
u
F
u
u u
ENERGYABSORBED
ENERGYDISSIPATED
ENERGYRECOVERED
TOTALENERGYDISSIPATED
Concept of Energy Absorbed and Dissipated
U.S. Army Corps of Engineers Structural Dynamics 33
-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
GR
OU
ND
AC
C,
gDevelopment of Effective
Earthquake Force Unlike wind loading,
earthquakes do not apply any direct forces on a structure
Earthquake ground motion causes the base to move, while masses at the floor levels try to stay in their places due to inertia.
This creates stresses in the resisting elements.
U.S. Army Corps of Engineers Structural Dynamics 34
utug ur
Development of Effective Earthquake Force
ug = Ground displacement
ur = Relative displacement
ut = Total displacement
= ug + ur
ut = Total acceleration
= ug + ur
:
: :
U.S. Army Corps of Engineers Structural Dynamics 35
m u t u t c u t k u tg r r r[ ( ) ( )] ( ) ( ) 0
mu t c u t k u t mu tr r r g ( ) ( ) ( ) ( )
Development of EffectiveEarthquake Force
Inertia force depends on the total acceleration of the masses.
Resisting forces from stiffness and damping depend on the relative displacement and velocity.
Thus, the equation of motion can be written as:
U.S. Army Corps of Engineers Structural Dynamics 36
Many ground motions now available via the Internet
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60
Time (sec)
Gro
un
d A
cce
lera
tio
n (
g's
)
-30
-20
-10
0
10
20
30
40
0 10 20 30 40 50 60
Time (sec)
Gro
un
d V
elo
city
(cm
/se
c)
-15
-10
-5
0
5
10
15
0 10 20 30 40 50 60
Time (sec)
Gro
un
d D
isp
lace
me
nt
(cm
)
Earthquake Ground Motion - 1940 El Centro
U.S. Army Corps of Engineers Structural Dynamics 37
)()()()( tumtuktuctum grrr
)()()()( tutum
ktu
m
ctu grrr
ξω2=m
c 2ω=m
k
Divide through by m:
Make substitutions:
)t(u)t(u)t(u)t(u grrr -=ω+ξω2+ 2
Simplified form:
“Simplified” form of Equation of Motion:
U.S. Army Corps of Engineers Structural Dynamics 38
For a given ground motion, the response history ur(t) is a function of the structure’s frequency w and
damping ratio x
)t(u)t(u)t(u)t(u grrr -=ω+ξω2+ 2
Ground motion acceleration history
Structural frequency
Damping ratio
“Simplified” form of Equation of Motion:
U.S. Army Corps of Engineers Structural Dynamics 39
Change in ground motion
or structural parameters x
and w requires re-calculation of structural response
-6
-4
-2
0
2
4
6
0 10 20 30 40 50 60
Time (sec)
Str
uct
ura
l D
isp
lace
me
nt
(in
)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60
Time (sec)
Gro
un
d A
cce
lera
tio
n (
g's
) Excitation applied to
structure with given x and w
Peak Displacement
Computed Response
SOLVER
Response to Ground Motion (1940 El Centro)
U.S. Army Corps of Engineers Structural Dynamics 40
0
4
8
12
16
0 2 4 6 8 10
PERIOD, Seconds
DIS
PL
AC
EM
EN
T, in
ches
5% Damped Response Spectrum for StructureResponding to 1940 El Centro Ground Motion
The Elastic Response Spectrum
An Elastic Response Spectrum is a plot of the peak computed relative displacement, ur, for an elastic structure with
a constant damping x and a varying fundamental frequency w (or period T=2p/w), responding to a given ground motion.
PE
AK
DIS
PL
AC
EM
EN
T, i
nc
he
s
U.S. Army Corps of Engineers Structural Dynamics 41
Computation of Deformation (or Displacement) Response Spectrum
U.S. Army Corps of Engineers Structural Dynamics 42
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, Seconds
Dis
pla
ce
me
nt,
Inc
he
s
Complete 5% Damped Elastic Displacement Response Spectrum for El Centro Ground Motion
U.S. Army Corps of Engineers Structural Dynamics 43
Development of PseudovelocityResponse Spectrum
D)T(PSV ω≡
5% Damping
U.S. Army Corps of Engineers Structural Dynamics 44
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 1.0 2.0 3.0 4.0
Period, Seconds
Ps
eu
do
ac
ce
lera
tio
n, i
n/s
ec
2
D)T(PSA 2ω≡
5% Damping
Development of PseudoaccelerationResponse Spectrum
U.S. Army Corps of Engineers Structural Dynamics 45
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.
5% Damping
Note about the Response Spectrum
U.S. Army Corps of Engineers Structural Dynamics 46
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
0.1 1 10Period (sec)
Ac
ce
lera
tio
n (
in/s
ec2 )
Total Acceleration Pseudo-Acceleration
Difference Between Pseudo-Acceleration and Total Acceleration
System with 5% Damping
U.S. Army Corps of Engineers Structural Dynamics 47
0.00
1.00
2.00
3.00
4.00
0.0 1.0 2.0 3.0 4.0 5.0
Period, Seconds
Ps
eu
do
ac
ce
lera
tio
n,
g
0%
5%
10%
20%
Damping
Pseudoacceleration Response Spectrafor Different Damping Values
U.S. Army Corps of Engineers Structural Dynamics 48
Example Structure
K = 500 kips/in.
W = 2,000 kips
M = 2000/386.4 = 5.18 kip-sec2/in.
w = (K/M)0.5 =9.82 rad/sec
T=2p/ w = 0.64 sec
5% Critical Damping
@T=0.64 sec, Pseudoacceleration = 301 in./sec2
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
400.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Period, Seconds
Pse
ud
oac
cele
rati
on
, in
/sec
2
Base Shear = M x PSA = 5.18(301) = 1559 kips
Use of an Elastic Response Spectrum
U.S. Army Corps of Engineers Structural Dynamics 49
0.10
1.00
10.00
100.00
0.01 0.10 1.00 10.00
PERIOD, Seconds
PS
EU
DO
VE
LOC
ITY,
in/s
ec
1.0
10.0
0.1
0.01
Acceleration, g
0.00
1
10.0
0.10
1.0
0.01
0.001
Displa
cem
ent,
in.
Four-Way Log Plot of Response Spectrum
U.S. Army Corps of Engineers Structural Dynamics 50
0.1
1
10
100
0.01 0.1 1 10Period, Seconds
Pse
ud
o V
elo
city
, In
/Sec
0% Damping
5% Damping
10% Damping
20* Damping
1.0
10.0
0.1
0.01
Acceleration, g
0.00
1
10.0
0.10
1.0
0.01
0.001
Displa
cem
ent,
in.
For a given earthquake,small variations in structural frequency (period) can producesignificantly different results.
1940 El Centro, 0.35 g, N-S
U.S. Army Corps of Engineers Structural Dynamics 51
0.1
1.0
10.0
100.0
0.01 0.10 1.00 10.00
Pse
uso
Ve
loci
ty,
in/s
ec
Period, seconds
El Centro
Loma Prieta
North Ridge
San Fernando
Average
Different earthquakeswill have different spectra.
5% Damped Spectra for Four California Earthquakes Scaled to 0.40 g (PGA)
U.S. Army Corps of Engineers Structural Dynamics 52
Relative Displacement
Total Acceleration
Ground Displacement
Zero
VERY FLEXIBLE STRUCTURE(T > 10 sec)
U.S. Army Corps of Engineers Structural Dynamics 53
Total Acceleration
Zero
Ground Acceleration
Relative Displacement
VERY STIFF STRUCTURE(T < 0.01 sec)
U.S. Army Corps of Engineers Structural Dynamics 54
ASCE 7-10 Uses a Smoothed Design Acceleration Spectrum
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6 7
Period, seconds
0.4SDS
Sa = SD1 / T
Sa = SDS(0.4 + 0.6 T/T0)
Sa = SD1 TL / T2
TST0
Sp
ectr
al A
ccel
era
tion,
g
SD1
SDS
TS = SD1 / SDS
T0 = 0.2TS
U.S. Army Corps of Engineers Structural Dynamics 55
Part II
Linear Multiple Degree of Freedom Systems
U.S. Army Corps of Engineers Structural Dynamics 56
Structural Dynamics of MDOF Systems
Uncoupling of Equations through use of Natural Mode Shapes
Solution of Uncoupled Equations
Recombination of Computed Response
Modal Response Spectrum Analysis
Equivalent Lateral Force Procedure
U.S. Army Corps of Engineers Structural Dynamics 57
Solving Equations of Motion
We need to develop a way to solve the equations of motion.
• This will be done by a transformation of coordinatesfrom Normal Coordinates (displacements at the nodes) To Modal Coordinates (amplitudes of the natural Mode shapes).
• Because of the Orthogonality Property of the natural mode shapes, the equations of motion become uncoupled, allowing them to be solved as SDOF equations.
• After solving, we can transform back to the normalcoordinates.
U.S. Army Corps of Engineers Structural Dynamics 58
Equations of Motion
Multi-degree-of-freedom system
- mass concentrated at floor levels which are subject to lateral displacements only
p(t)ukucum rrr
U.S. Army Corps of Engineers Structural Dynamics 59
Equations of Motion
Free vibration
[C] = [0], {p(t)} = {0}
0 rr ukum
U.S. Army Corps of Engineers Structural Dynamics 60
Equations of Motion
Motion of a system in free vibration is simple harmonic
{ } { }
{ } { }
{ } { } tsinAu
tcosA u
tsinAu
r
r
r
ωω-=
ωω=
ω=
2
U.S. Army Corps of Engineers Structural Dynamics 61
Equations of Motion
[ ]{ } [ ]{ } { }
[ ]{ } { }0=ω-
0=+ω2
2
Amk
AkAm-
U.S. Army Corps of Engineers Structural Dynamics 62
Example
Given
hs = 10 ft
w = 386.4 kips/floor
E = 4000 ksi
Icol = 4500 in.4 each column (0.7Ig )
U.S. Army Corps of Engineers Structural Dynamics 63
Determine Mass Matrix
m = w/g = 386.4 / 386.4
= 1.0 kip-sec.2/in.
100
010
001
m
U.S. Army Corps of Engineers Structural Dynamics 64
Determine Stiffness Matrix
k = 12EI/hs3 = 12 x 4000 x 9000 / (12x10)3 = 250 kips/in.
kij = force corresponding to displacement of coordinate i resulting from a unit displacement of coordinate j
110
121
012
250k
250
250
250
K13 = 0
K23 = -250
K33 = 250
K12 = -250
K22 = 500
K32 = -250
K11 = 500
K21 = -250
K31 = 0
1
1
1
U.S. Army Corps of Engineers Structural Dynamics 65
Find Determinant for Matrix [k] - 2[m]
The period is equal to 2/:
T1 = 0.893 sec.
T2 = 0.319 sec.
T3 = 0.221 sec.
Setting the determinant of the above matrix equal to zero yields the following frequencies:
2
2
2
2
2502500
250500250
0250500
mk
1 = 7.036 radians/sec.
2 = 19.685 radians/sec.
3 = 28.491 radians/sec.
U.S. Army Corps of Engineers Structural Dynamics 66
Find Mode Shapes
First Mode:
0
0
0
)036.7(2502500
250)036.7(500250
0250)036.7(500
11
21
31
2
2
2
31 = 1.0, 21 = 0.802, 11 = 0.445
U.S. Army Corps of Engineers Structural Dynamics 67
Second Mode:
32 = 1.0, 22 = -0.55, 12 = -1.22
Find Mode Shapes
0
0
0
)685.19(2502500
250)685.19(500250
0250)685.19(500
12
22
32
2
2
2
U.S. Army Corps of Engineers Structural Dynamics 68
Third Mode:
33 = 1.0, 23 = -2.25, 13 = 1.802
Find Mode Shapes
0
0
0
)491.28(2502500
250)491.28(500250
0250)491.28(500
13
23
33
2
2
2
U.S. Army Corps of Engineers Structural Dynamics 69
Orthogonality Properties of Natural Mode Shapes
The natural modes of vibration of any multi degree of freedom system are orthogonal with respect to the mass and stiffness matrices. The same type of orthogonality can be assumed to apply to the damping matrix as well:
{m}T[m] {n} = 0
{m}T[c] {n} = 0 for m ≠ n
{m}T[k] {n} = 0
U.S. Army Corps of Engineers Structural Dynamics 70
Mode Superposition Analysis of Earthquake Response
3
2
1
333231
232221
131211
3
2
1
X
X
X
u
u
u
r
r
r
3332321313
3232221212
3132121111
XXXu
XXXu
XXXu
r
r
r
or
or
XXur 321
U.S. Army Corps of Engineers Structural Dynamics 71
Mode Superposition Analysis of Earthquake Response
U.S. Army Corps of Engineers Structural Dynamics 72
Modal Response Spectrum Analysis
An elastic dynamic analysis of structure utilizing the peak dynamic response of all modes having a significant contribution to total structural response. Peak modal responses are calculated using the ordinates of the appropriate response spectrum curve which correspond to the modal periods. Maximum modal contributions are combined in a statistical manner to obtain an approximate total structural response.
U.S. Army Corps of Engineers Structural Dynamics 73
ASCE 7-10 Section 11.4.5General Procedure Design Spectrum
A five percent damped elastic design response
spectrum constructed in accordance with ASCE 7-10
Figure 11.4-1, using the values of SDS and SD1
consistent with the specific site.
U.S. Army Corps of Engineers Structural Dynamics 74
ASCE 7-10 Section 11.4.7Site-Specific Ground Motion Procedures
The site-specific ground motion procedures set forth in Chapter 21 are permitted to be used to determine ground motions for any structure.
U.S. Army Corps of Engineers Structural Dynamics 75
Response Spectrum Analysis
For each mode m, determine:
Earthquake participation factor
Modal Mass
1
2
i
imim g
wM
n
i
imim g
wL1
U.S. Army Corps of Engineers Structural Dynamics 76
Determine Modal Mass and Participation Factors for Each Mode
= 1.0 kip-sec2/in. (f11 + f21 + f31)
= 1.0 (0.445 + 0.802 + 1.0)
= 2.247 kip-sec2/in.
g
wL i
ii
3
11
1
= 1.0 kip-sec2/in. (f112 + f21
2 + f31
2)
= 1.0 (0.4452 + 0.8022 + 1.02)
= 1.841 kip-sec2/in.
g
wM i
ii
3
1
21
1
U.S. Army Corps of Engineers Structural Dynamics 77
Determine Modal Mass and Participation Factors for Each Mode
= 1.0 kip-sec2/in. (f12 + f22 + f32)
= 1.0 (-1.22 - 0.55 + 1.0)
= -0.77 kip-sec2/in.
g
wL i
ii
3
12
2
= 1.0 kip-sec2/in. (f122 + f22
2 + f32
2)
= 1.0 (1.222 + 0.552 + 1.02)
= 2.791 kip-sec2/in.
g
wM i
ii
3
1
22
2
U.S. Army Corps of Engineers Structural Dynamics 78
Determine Modal Mass and Participation Factors for Each Mode
= 1.0 kip-sec2/in. (f13 + f23 + f33)
= 1.0 (1.802 - 2.25 + 1.0)
= 0.552 kip-sec2/in.
g
wL i
ii
3
13
3
= 1.0 kip-sec2/in. (f132 + f23
2 + f33
2)
= 1.0 (1.8022 + 2.252 + 1.02)
= 9.310 kip-sec2/in.
g
wM i
ii
3
1
23
3
U.S. Army Corps of Engineers Structural Dynamics 79
Response Spectrum Analysis
For each mode m, determine (cont’d):
Effective Weight
Participating Mass
gM
LW
m
mm
2
W
WPM m
n
iiwW
1
Where weight at floor level i
U.S. Army Corps of Engineers Structural Dynamics 80
Determine Effective Weight and Participation Mass for Each Mode
kipsin
in
kipg
M
LW 72.1059
sec
.
.
sec4.386
841.1
247.22
22
1
21
1
≈ 3 x w = 1159.2 kips
kipsgM
LW 65.124.386
310.9
)552.0( 2
3
23
3
kipsgM
LW 08.824.386
791.2
)77.0( 2
2
22
2
kipsWi 45.1154
U.S. Army Corps of Engineers Structural Dynamics 81
Determine Effective Weight and Participation Mass for Each Mode
996.0
011.04.3863
65.12
071.04.3863
08.82
914.04.3863
72.1059
33
22
11
PM
W
WPM
W
WPM
W
WPM
U.S. Army Corps of Engineers Structural Dynamics 82
Response Spectrum Analysis
Determine number of modes to be considered... to
represent at least 90% of participating mass of
structure (ASCE 7-05 Sec. 12.9.1)
PM = (Wm/W) 0.90
U.S. Army Corps of Engineers Structural Dynamics 83
Response Spectrum Analysis
Determine spectral acceleration for each mode from design response spectra
Mode 1: T1 = 0.893 sec → Sa1 = 0.084g
Mode 2: T2 = 0.319 sec → Sa2 = 0.24g
Mode 3: T3 = 0.221 sec → Sa3 = 0.34g
U.S. Army Corps of Engineers Structural Dynamics 84
Response Spectrum Analysis
Determine base shear for each mode
Mode 1: V1 = 0.0840 x 1059.72 = 89.0 kips
Mode 2: V2 = 0.24 x 82.08 = 19.7 kips
Mode 3: V3 = 0.34 x 12.65 = 4.3 kips
U.S. Army Corps of Engineers Structural Dynamics 85
Response Spectrum Analysis
Distribute base shear for each mode over height of structure
where Fim = lateral force at level i for mode m
Vm = base shear for mode m
mimi
imiim V
w
wF
U.S. Army Corps of Engineers Structural Dynamics 86
Distribute Base Shear for Each Mode over Height of Structure
Level, i Weight, wi fi1 wi fi1 Fi1
3 386.4 1 386.4 39.6
2 386.4 0.802 309.9 31.7
1 386.4 0.445 171.9 17.6
= 868.2 88.9
Mode 1 V1 = 89.0 kips
U.S. Army Corps of Engineers Structural Dynamics 87
Distribute Base Shear for Each Mode over Height of Structure
Level, i Weight, wi fi2 wi fi2 Fi2
3 386.4 1 386.4 -25.2
2 386.4 -0.55 -212.5 14.1
1 386.4 -1.22 -471.4 31.2
= -297.5 20.1
Mode 2 V2 = 19.7 kips
U.S. Army Corps of Engineers Structural Dynamics 88
Distribute Base Shear for Each Mode over Height of Structure
Level, i Weight, wi fi3 wi fi3 Fi3
3 386.4 1 386.4 7.8
2 386.4 -2.25 -869.4 -17.6
1 386.4 1.802 696.3 14.1
= 213.3 4.3
Mode 3 V3 = 4.3 kips
U.S. Army Corps of Engineers Structural Dynamics 89
Response Spectrum Analysis Example
39.6
31.7 17.6
17.6
25.2
14.1
14.131.2
7.8
Mode 1 Mode 2 Mode 3
U.S. Army Corps of Engineers Structural Dynamics 90
Response Spectrum Analysis
Perform lateral analysis for each mode... to determine member forces for each mode of vibration being considered
U.S. Army Corps of Engineers Structural Dynamics 91
Response Spectrum Analysis
Combine dynamic analysis results (moments, shears,
axial forces, and displacements) for all considered
modes using root mean square combination (SRSS)...
to approximate total structural response or resultant
design values
U.S. Army Corps of Engineers Structural Dynamics 92
ASCE 7 Allows an Approximate Modal Analysis Technique Called the “EQUIVALENT LATERAL FORCE PROCEDURE”
• Empirical Period of Vibration
• Smoothed Response Spectrum
• Compute Total Base Shear V as if SDOF
• Distribute V Along Height assuming “Regular” Geometry
• Compute Displacements and Member Forces using Standard Procedures
ASCE 7 - Approximate Modal Analysis
U.S. Army Corps of Engineers Structural Dynamics 93
Equivalent Lateral Force Procedure
Method is based on FIRST MODE response
Higher modes can be included empirically
Has been calibrated to provide a reasonable estimate of the envelopeof story shear, NOT to provide accurate estimates of story force
May result in overestimate of overturning moment. ASCE 7 compensates.
U.S. Army Corps of Engineers Structural Dynamics 94
Assume first mode effective mass = Total Mass = M = W/g
Use Response Spectrum to obtain Total Acceleration @ T1
T1
Sa1/g
Period, sec
Acceleration, g
WSg
WgSMgSV aaaB 111 )()(
Equivalent Lateral Force Procedure
U.S. Army Corps of Engineers Structural Dynamics 95
1st
Mode 2nd
Mode Combined
+ =
Equivalent Lateral Force ProcedureHigher Mode Effects
U.S. Army Corps of Engineers Structural Dynamics 96
VCF vxx
n
i
kii
kxx
vx
hw
hwC
1
Distribution of Forces along Height
U.S. Army Corps of Engineers Structural Dynamics 97
k=1 k=2
0.5 2.5
2.0
1.0
Period, Sec
k
k = 0.5T + 0.75(sloped portion only)
k accounts for Higher Mode Effects
U.S. Army Corps of Engineers Structural Dynamics 98
Thank You!!
Any Remaining Questions?