# Equations of motion for SDOF structures

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Structural Dynamics of                                                                                                Structural Dynamics
Linear Elastic Single-Degree-of-Freedom
(SDOF) Systems                                                                                 • Equations of motion for SDOF structures
• Structural frequency and period of vibration
• Dynamic magnification and resonance
• Effect of damping on behavior
• Linear elastic response spectra

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 1                     Instructional Material Complementing FEMA 451, Design Examples              SDOF Dynamics 3 - 2

Importance in Relation to ASCE 7-05
Idealized SDOF Structure
• Ground motion maps provide ground
accelerations in terms of response spectrum
F(t)
coordinates.                                                                                                                                  F ( t ), u ( t )
Mass
• Equivalent lateral force procedure gives base
shear in terms of design spectrum and period                                                                                                                                                                                  t
Damping
of vibration.
Stiffness
• Response spectrum is based on 5% critical                                                                                                                                  u(t)
damping in system.
• Modal superposition analysis uses design
response spectrum as basic ground motion                                                                                                                                                                                      t
input.

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 3                     Instructional Material Complementing FEMA 451, Design Examples              SDOF Dynamics 3 - 4

Equation of Dynamic Equilibrium                                                                          Observed Response of Linear SDOF
Applied Force, kips
40

f I (t )                       F (t )                                      0

-40
0.00             0.20           0.40           0.60           0.80          1.00

fD (t )                                                                        Displacement, in
0.50
0 .5 f S ( t )                                                 0 .5 f S ( t )
0.00

-0.50
0.00             0.20            0.40           0.60           0.80          1.00

Velocity, in/sec
15.00

0.00

-15.00

F (t ) − f I (t ) − fD (t ) − fS (t ) = 0
0.00             0.20           0.40           0.60           0.80          1.00

Acceleration, in/sec2
400.00

fI (t ) + fD (t ) + fS (t ) = F (t )                                                                        0.00

-400.00
0.00             0.20           0.40           0.60           0.80          1.00

Time, sec
Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 5                     Instructional Material Complementing FEMA 451, Design Examples              SDOF Dynamics 3 - 6

FEMA 451B Topic 3 Handouts                                                                                                                                                                 Structural Dynamics 1
Observed Response of Linear SDOF                                                                                                                                          Equation of Dynamic Equilibrium
(Development of Equilibrium Equation)                                                                                                                                                                         f I (t )                                       F (t )
Spring Force, kips                          Damping Force, Kips                                                      Inertial Force, kips
30.00                                        4.00                                                                    50.00

fD (t )
15.00                                        2.00                                                                    25.00
0 .5 f S ( t )                                                                  0 .5 f S ( t )
0.00                                        0.00                                                                     0.00

-15.00                                       -2.00                                                                   -25.00

-30.00                                       -4.00                                                                   -50.00
-0.60    -0.30   0.00    0.30    0.60       -20.00    -10.00    0.00            10.00                   20.00         -500       -250       0     250     500
Displacement, inches                             Velocity, In/sec                                                       Acceleration, in/sec2

Slope = k                                     Slope = c                                                            Slope = m                                             fI (t ) + fD (t ) + fS (t ) = F (t )
= 50 kip/in                               = 0.254 kip-sec/in                                                   = 0.130 kip-sec2/in

f S ( t ) = k u( t )                             f D (t ) = c u t )
&(                                                         f I (t ) = m u t )
&&(                                     m u( t ) + c u( t ) + k u( t ) = F ( t )
&&         &

Instructional Material Complementing FEMA 451, Design Examples                                                 SDOF Dynamics 3 - 7                             Instructional Material Complementing FEMA 451, Design Examples                               SDOF Dynamics 3 - 8

Properties of Structural Mass                                                                                                                                   Properties of Structural Damping
Mass
Internal Force

Damping Force
Damping

M                                                                                                                                             C
1.0                                                                                                                                              1.0

Acceleration                                                                                                                                           Velocity
• Includes all dead weight of structure                                                                                                                • In absence of dampers, is called inherent damping
• May include some live load                                                                                                                           • Usually represented by linear viscous dashpot
• Has units of force/acceleration                                                                                                                      • Has units of force/velocity

Instructional Material Complementing FEMA 451, Design Examples                                                 SDOF Dynamics 3 - 9                             Instructional Material Complementing FEMA 451, Design Examples                              SDOF Dynamics 3 - 10

Properties of Structural Damping (2)                                                                                                                                    Properties of Structural Stiffness
Spring Force
Damping Force

Damping
Stiffness

K
1.0

AREA =
ENERGY                                                                                                                                                                                                                  Displacement
DISSIPATED                                                        Displacement
•   Includes all structural members
Damping vs displacement response is                                                                                                                     •   May include some “seismically nonstructural” members
elliptical for linear viscous damper.                                                                                                                   •   Requires careful mathematical modelling
•   Has units of force/displacement

Instructional Material Complementing FEMA 451, Design Examples                                                SDOF Dynamics 3 - 11                             Instructional Material Complementing FEMA 451, Design Examples                              SDOF Dynamics 3 - 12

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                     Structural Dynamics 2
Properties of Structural Stiffness (2)                                                                                                  Undamped Free Vibration

Equation of motion:              m u( t ) + k u( t ) = 0
&&

Spring Force
Initial conditions:               &
u0 u0
Stiffness

Assume:      u ( t ) = A sin( ω t ) + B cos( ω t )
&
u0                                                                  k
Solution:     A=                                 B = u0                          ω =
ω
AREA =
ENERGY
DISSIPATED                        Displacement
m
• Is almost always nonlinear in real seismic response                                                                                            &
u0
• Nonlinearity is implicitly handled by codes                                                                                     u (t ) =                sin( ω t ) + u 0 cos( ω t )
• Explicit modelling of nonlinear effects is possible                                                                                             ω
Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 13                         Instructional Material Complementing FEMA 451, Design Examples    SDOF Dynamics 3 - 14

Undamped Free Vibration (2)                                                                                      Approximate Periods of Vibration
(ASCE 7-05)
&
u0            T = 0.5 sec

Ta = Ct hnx
1.0
Displacement, inches

3
2
1
0
u0                                                                                                                    Ct =   0.028, x = 0.8           for steel moment frames
-1                                                                                                                                Ct =   0.016, x = 0.9           for concrete moment frames
-2
-3
Ct =   0.030, x = 0.75          for eccentrically braced frames
0.0                         0.5                            1.0                         1.5                           2.0    Ct =   0.020, x = 0.75          for all other systems
Note: This applies ONLY to building structures!
Time, seconds

Circular Frequency
Cyclic Frequency
(cycles/sec, Hertz)
Period of Vibration
(sec/cycle)                                                    T = 0.1N
a
k                                ω                                  1   2π
ω =                                                   f =                                T =   =                                       For moment frames < 12 stories in height, minimum
m                                2π                                 f   ω                                     story height of 10 feet. N = number of stories.

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 15                         Instructional Material Complementing FEMA 451, Design Examples    SDOF Dynamics 3 - 16

Empirical Data for Determination
of Approximate Period for Steel Moment Frames                                                                        Periods of Vibration of Common Structures
20-story moment resisting frame                                           T = 1.9 sec
10-story moment resisting frame                                           T = 1.1 sec
1-story moment resisting frame                                            T = 0.15 sec

20-story braced frame                                                     T = 1.3 sec
Ta = 0.028h          0.8                   10-story braced frame                                                     T = 0.8 sec
n
1-story braced frame                                                      T = 0.1 sec

Gravity dam                                                               T = 0.2 sec
Suspension bridge                                                         T = 20 sec

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 17                         Instructional Material Complementing FEMA 451, Design Examples    SDOF Dynamics 3 - 18

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                            Structural Dynamics 3
Adjustment Factor on Approximate Period                                                                                                                                 Which Period of Vibration to Use
(Table 12.8-1 of ASCE 7-05)                                                                                                                       in ELF Analysis?

T = Ta Cu ≤ Tcomputed                                                                                                If you do not have a “more accurate” period
(from a computer analysis), you must use T = Ta.
SD1                                                                     Cu                                   If you have a more accurate period from a computer
> 0.40g                                                                    1.4                                  analysis (call this Tc), then:
0.30g                                                                    1.4
0.20g                                                                    1.5                                                                     if Tc > CuTa                            use T = CuTa
0.15g                                                                    1.6
< 0.1g                                                                    1.7                                                                     if Ta < Tc < TuCa use T = Tc
Applicable ONLY if Tcomputed comes from a “properly
substantiated analysis.”                                                                                                                                                if Tc < Ta                              use T = Ta

Instructional Material Complementing FEMA 451, Design Examples                  SDOF Dynamics 3 - 19                                                     Instructional Material Complementing FEMA 451, Design Examples           SDOF Dynamics 3 - 20

Damped Free Vibration                                                                                                                                      Damping in Structures
c    c
Equation of motion:  m u( t ) + c u( t ) + k u( t ) = 0
&&      &                                                                                                                    ξ =     =                                           cc is the critical damping constant.
2mω   cc
Initial conditions: u0       &
u0
ξ is expressed as a ratio (0.0 < ξ < 1.0) in computations.
Assume: u ( t ) = e
st

Sometimes ξ is expressed as a% (0 < ξ < 100%).
Solution:
Displacement, in
⎡                   u + ξω u 0
&                      ⎤
u ( t ) = e − ξω t ⎢ u 0 cos( ω D t ) + 0         sin( ω D t ) ⎥
⎣                      ωD                   ⎦
c    c                                                                                                                                                                                                                          Time, sec
ξ =             =                                          ωD = ω 1− ξ2
2mω   cc
Response of Critically Damped System, ξ=1.0 or 100% critical

Instructional Material Complementing FEMA 451, Design Examples                  SDOF Dynamics 3 - 21                                                     Instructional Material Complementing FEMA 451, Design Examples           SDOF Dynamics 3 - 22

Damping in Structures
Damped Free Vibration (2)
True damping in structures is NOT viscous. However, for low
damping values, viscous damping allows for linear equations
and vastly simplifies the solution.
Displacement, inches

3
2
Spring Force, kips                          Damping Force, Kips                             Inertial Force, kips                                        1                                                                                                   0% Damping
30.00                                        4.00                                           50.00                                                                0                                                                                                   10% Damping
-1                                                                                                   20% Damping
15.00                                        2.00                                           25.00
-2
-3
0.00                                        0.00                                            0.00
0.0             0.5                   1.0                   1.5                    2.0
-15.00                                       -2.00                                          -25.00                                                                                                Time, seconds

-30.00                                       -4.00                                          -50.00
-0.60   -0.30   0.00     0.30    0.60       -20.00   -10.00   0.00     10.00   20.00         -500     -250     0       250    500
Displacement, inches                             Velocity, In/sec                             Acceleration, in/sec2

Instructional Material Complementing FEMA 451, Design Examples                  SDOF Dynamics 3 - 23                                                     Instructional Material Complementing FEMA 451, Design Examples           SDOF Dynamics 3 - 24

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                         Structural Dynamics 4
Damping in Structures (2)                                                                                                                Damping in Structures (3)
Welded steel frame                                                            ξ = 0.010                                                                     Inherent damping
Bolted steel frame                                                            ξ = 0.020                                                                   ξ      is a structural (material) property
independent of mass and stiffness
Uncracked prestressed concrete                                                ξ = 0.015
Uncracked reinforced concrete                                                 ξ = 0.020                                                                  ξ Inherent = 0.5 to 7.0% critical
Cracked reinforced concrete                                                   ξ = 0.035

Glued plywood shear wall                                                      ξ = 0.100
Nailed plywood shear wall                                                     ξ = 0.150
ξ     is a structural property dependent on
Damaged steel structure                                                       ξ = 0.050                                              C                          mass and stiffness and
Damaged concrete structure                                                    ξ = 0.075                                                                         damping constant C of device

Structure with added damping                                                  ξ = 0.250                                                                    ξ Added = 10 to 30% critical

Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 25                                       Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 26

For all
1
damping values            Equation of motion:                               m u&( t ) + k u ( t ) = p 0 sin( ω t )
&
u            2 πξ                       ω
u1                                                                   ln 1 =                                              = frequency of the forcing function
1− ξ2                                  2π
0.5
u2                                                    u2
u3
T =
Amplitude

0                                                                                                                                      ω                                       T     = 0.25 sec
po=100 kips
For very low
− ξω t                                                                           150
-0.5                                  u0 e                                            damping values                          100
Force, Kips

50
u1 − u 2                            0
ξ≅                                      -50
2π u2
-1
-100
0.00     0.50        1.00         1.50       2.00         2.50        3.00
-150
Time, Seconds                                                                                     0.00      0.25       0.50         0.75         1.00        1.25         1.50       1.75       2.00
Time, Seconds

Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 27                                       Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 28

Equation of motion: m u ( t ) + k u ( t ) = p 0 s in ( ω t )
Define                        β =
Assume system is initially at rest:                                                                                                                                        ω            Structure’s natural frequency

Particular solution: u ( t ) = C s i n ( ω t )                                                                                                                 Dynamic magnifier
Transient response
(at structure’s frequency)
Complimentary solution: u( t ) = A sin(ωt ) + B cos(ωt )
p0  1
u( t ) =                          (sin(ω t ) − β sin(ω t ))
Solution:                                                                                                                                                  k 1− β2
p0      1       ⎛            ω          ⎞
u (t ) =                    2 ⎜
sin(ω t ) − sin(ω t ) ⎟                                                                                                                     Steady state
k 1 − (ω / ω ) ⎝             ω          ⎠                                                                         Static displacement                          response

Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 29                                       Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 30

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                   Structural Dynamics 5
ω = 4π rad / sec ω = 2π rad / sec                                                                          β = 0.5 uS = 5.0 in.                         ω ≈ 4 π rad / sec ω = 4 π rad / sec                                                                                           β = 0.99 u S = 5.0 in.
150
200
100
100                                                                                                                                                                             50
-100                                                                                                                                                                           -5 0
(kips)                                          -1 0 0
-200
-1 5 0
0.00           0.25          0.50       0.75          1.00                 1.25      1.50       1.75        2.00                                                                0 .0 0        0 .2 5      0 .5 0    0 .7 5           1 .0 0           1 .2 5    1 .5 0    1 .7 5       2 .0 0

10                                                                                                                                                                           5 00

response (in.)                 0                                                                                                                          response (in.)                                       0
-5                                                                                                                                                                          -2 50
-10                                                                                                                                                                          -5 00
0.00           0.25          0.50       0.75          1.00                 1.25      1.50       1.75        2.00                                                               0.00           0 .25        0 .5 0    0 .7 5           1 .0 0           1.2 5     1.5 0     1.75        2.00

10                                                                                                                                                                           500

5                                                                                                                                                                           250
Transient                      0                                                                                                                                                                                0
Transient
response (in.)                -5
response (in.)                                  -250
-10                                                                                                                                                                          -500
0.00           0.25          0.50       0.75          1.00                 1.25      1.50       1.75        2.00                                                              0 .00           0.2 5        0.50      0 .75            1.0 0            1.25      1 .50     1.7 5       2.00

10                                                                                                                                                                              80
,

5

,
40
Total response                 0                                                                                                                          Total response                                        0
(in.)                         -5                                                                                                                          (in.)                                             -40

p
p

-10
-80
0.00           0.25          0.50       0.75          1.00                 1.25      1.50       1.75        2.00                                                               0.00            0.25        0.50      0.75             1.00             1.25      1.50      1.75        2.00
Time, seconds                                                                                                                                                                T ime, seconds
Instructional Material Complementing FEMA 451, Design Examples                            SDOF Dynamics 3 - 31                                                                  Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 32

ω ≈ 4π rad / sec ω = 4π rad / sec                                                                                            β = 1.01           u S = 5.0 in.
Undamped Resonant Response Curve                                                                                                                                                     150
100
50
-5 0
-1 0 0
-1 5 0

40
2π uS                                                       500
0 .0 0        0 .2 5       0 .5 0    0 .7 5           1 .0 0          1 .2 5    1 .5 0    1 .7 5       2 .0 0
Displacement, in.

response (in.)                                         0
-2 5 0
-5 0 0
0                                                                                                                                                                                        0 .0 0        0 .2 5       0 .5 0    0 .7 5           1 .0 0          1 .2 5    1 .5 0    1 .7 5      2 .0 0

500
250
Transient                                              0
-40                                                                                                                                      response (in.)                                   -2 5 0
-5 0 0
Linear envelope                                                                                                                                      0 .0 0        0 .2 5       0 .5 0    0 .7 5          1 .0 0           1 .2 5    1 .5 0   1 .7 5       2 .0 0

80
,

-80                                                                                                                                      Total response
40

0
0.00    0.25            0.50            0.75        1.00          1.25                  1.50        1.75          2.00                (in.)                                                -4 0
p

Time, seconds                                                                                                                                    -8 0
0 .0 0       0 .2 5      0 .5 0   0 .7 5           1 .0 0           1.2 5    1.5 0     1 .7 5      2 .0 0
T im e , s ec o n d s
Instructional Material Complementing FEMA 451, Design Examples                            SDOF Dynamics 3 - 33                                                                  Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 34

ω = 4π rad / sec                                  ω =8π rad / sec                                          β = 2.0 uS = 5.0 in.                                                                 Response Ratio: Steady State to Static
150
(Signs Retained)
100
12.00
0
-50
-100                                                                                                                                                          8.00
In phase
Magnification Factor 1/(1-β 2)

-150
0.00          0.25        0.50       0.75             1.00              1.25      1.50        1.75        2.00
6
3
response (in.)                     0                                                                                                                                                                                                               Resonance
-3
-6
0.00
0 .0 0        0 .2 5      0 .5 0     0.7 5            1 .00             1 .2 5    1 .5 0      1 .7 5      2 .0 0

6
3
-4.00
Transient response                  0

(in.)                              -3
-6
-8.00                                                    180 degrees out of phase
0 .0 0      0 .2 5      0 .5 0     0 .7 5           1 .0 0            1 .2 5    1 .5 0      1 .75      2.00

6
-12.00
,

3
Total response                      0                                                                                                                                                             0.00              0.50               1.00         1.50                      2.00            2.50           3.00
(in.)                              -3
Frequency Ratio β
p

-6
0 .0 0      0 .2 5      0 .5 0     0 .7 5           1 .0 0            1 .2 5    1 .5 0      1 .7 5     2 .0 0
T im e , s e c o n d s
Instructional Material Complementing FEMA 451, Design Examples                            SDOF Dynamics 3 - 35                                                                  Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 36

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                   Structural Dynamics 6
(Absolute Values)
12.00                                                                                                                                                     Equation of motion:

10.00
Resonance
m u t ) + cu t ) + k u ( t ) = p 0 sin(ω t )
&&(      &(
Magnification Factor 1/(1-β 2)

2π
8.00
T =             = 0.25 sec
ω
6.00
150                                                                                                                         po=100 kips
Slowly                                                                                                                   100

Force, Kips
4.00                                                                                                                                                             50

2.00
Rapidly                                               -50
-100
1.00
0.00                                                                                                                                                               0.00                                     0.25         0.50         0.75         1.00         1.25        1.50    1.75       2.00
0.00                                          0.50            1.00            1.50           2.00             2.50          3.00                                                                                                         Time, Seconds
Frequency Ratio β
Instructional Material Complementing FEMA 451, Design Examples            SDOF Dynamics 3 - 37                                                                     Instructional Material Complementing FEMA 451, Design Examples      SDOF Dynamics 3 - 38

Equation of motion:
m u ( t ) + cu ( t ) + k u ( t ) = p 0 sin(ω t )
&&         &                                                                                                                                                                                                                       Transient response at structure’s frequency
(eventually damps out)

Assume system is initially at rest                                                                                                                                                                                    u ( t ) = e − ξω t [ A sin(ω D t ) + B cos(ω D t ) ] +
Particular solution:                                                             u ( t ) = C sin(ω t ) + D cos(ω t )
C sin(ω t ) + D cos(ω t )
Complimentary solution:
[ A sin(ω D t ) + B cos(ω D t ) ]
− ξω t                                                                                                                                                                                Steady state response,
u(t ) = e                                                                                                                                           c
2 mω
Solution:
po      1− β 2                                          po      − 2ξβ
u ( t ) = e − ξω t [ A sin(ω D t ) + B cos(ω D t ) ]                                                                                       ωD = ω 1− ξ2                                                            C=                                                        D=
k (1 − β ) + (2ξβ ) 2
2 2
k (1 − β 2 ) 2 + (2ξβ ) 2
+ C sin(ω t ) + D cos(ω t )

Instructional Material Complementing FEMA 451, Design Examples            SDOF Dynamics 3 - 39                                                                     Instructional Material Complementing FEMA 451, Design Examples      SDOF Dynamics 3 - 40

BETA=1 (Resonance)
50
Beta=0.5
Beta=2.0                                                                                                            40
Displacement Amplitude, Inches

50
1
δ Static
30

2ξ
40
Displacement Amplitude, Inches

20
30
10
20                                                                                                                                                              0
10                                                                                                                                                             -10
0                                                                                                                                                             -20
-10                                                                                                                                                            -30
-20                                                                                                                                                            -40
-30                                                                                                                                                            -50
-40                                                                                                                                                               0.00             1.00             2.00              3.00              4.00        5.00
Time, Seconds
-50
0.00                1.00              2.00               3.00                4.00            5.00
Time, Seconds

Instructional Material Complementing FEMA 451, Design Examples            SDOF Dynamics 3 - 41                                                                     Instructional Material Complementing FEMA 451, Design Examples      SDOF Dynamics 3 - 42

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                         Structural Dynamics 7
Resonance
Effects of Damping
12.00                                                                0.0% Damping
5.0 % Damping
200
10.0% Damping

Dynamic Response Amplifier
Displacement Amplitude, Inches

150                                                                                                                                    10.00                                                                25.0 % Damping

100
8.00
1
RD =
50

0
6.00                                                         (1 − β ) + ( 2ξβ ) 2
2 2
-50

-100                                                                                                                                    4.00
-150                                                                                                                                            Slowly
-200                                                                                                                                    2.00                                                                           Rapidly
0.00             1.00              2.00             3.00                4.00       5.00                                                                                                                            loaded
Time, Seconds                                                                                   0.00
0.00        0.50           1.00            1.50           2.00             2.50       3.00
0% Damping            %5 Damping
Frequency Ratio, β

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 43                                               Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 44

Summary Regarding Viscous Damping                                                                                                                  Summary Regarding Viscous Damping
• For systems loaded at a frequency near their
• Damping is an effective means for dissipating energy in
natural frequency, the dynamic response                                                                                                                            the system. Unlike strain energy, which is recoverable,
exceeds the static response. This is referred to                                                                                                                   dissipated energy is not recoverable.
as dynamic amplification.
• A damped system, loaded at resonance, will have a
• An undamped system, loaded at resonance, will                                                                                                                     limited displacement over time with the limit being (1/2ξ)
have an unbounded increase in displacement                                                                                                                         times the static displacement.
over time.                                                                                                                                    • Damping is most effective for systems loaded at or near
resonance.

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 45                                               Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 46

CONCEPT of ENERGY STORED
and Energy DISSIPATED
Energy                                                                                        Energy
F      Stored                                                                F                       Dissipated
2
1                                                               1                                                F(t)
u                                                                u
Total
Energy                                                                                            Energy
F       Recovered
F                 Dissipated
3                                                                                                      Time, T
2

u                                                                             4
3                                               u

Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 47                                               Instructional Material Complementing FEMA 451, Design Examples          SDOF Dynamics 3 - 48

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                  Structural Dynamics 8
Piecewise Exact Method
Solution Techniques

• Fourier transform
dF
• Duhamel integration                                                                                                                            F (τ ) = Fo + τ
Fo
dt
• Piecewise exact
• Newmark techniques                                                                                                                                 dF

dt
All techniques are carried out numerically.                                                                                                                             τ
dt
Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 49              Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 50

Piecewise Exact Method
Piecewise Exact Method
Initial conditions        uo ,0 = 0 uo ,0 = 0
&
Determine “exact” solution for 1st time step
• Exact if load increment is linear
u1 = u (τ )                   u1 = u (τ )
&    &                            u1 = u (τ )
&& &&                           • Very computationally efficient
Establish new initial conditions
u o ,1 = u (τ )                       u 0,1 = u (τ )
&       &                                         • Not generally applicable for inelastic behavior
LOOP
Obtain exact solution for next time step                                                            Note: NONLIN uses the piecewise exact method for
u 2 = u (τ )                 u 2 = u (τ )
&     &                             u 2 = u (τ )
&& &&                          response spectrum calculations.

Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 51              Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 52

Newmark Techniques                                                                                     Newmark Method

• Proposed by Nathan Newmark                                                                            • Works for inelastic response
• General method that encompasses a family of different
• Derived by:
– Development of incremental equations of motion
• Potential numerical error
– Assuming acceleration response over short time step
Note: NONLIN uses the Newmark method for
general response history calculations

Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 53              Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 54

FEMA 451B Topic 3 Handouts                                                                                                                                                   Structural Dynamics 9
Development of Effective Earthquake Force                                                                                                                                                              Earthquake Ground Motion, 1940 El Centro
0.4

0.3

Ground Acceleration (g's)
0.2

0.1
40
0
30

Ground Velocity (cm/sec)
-0.1
20
-0.2
10
-0.3
0             10        20       30        40       50                                           60                                0
Time (sec)
-10
15
-20

Ground Displacement (cm)
10
-30
0        10      20        30         40        50        60
5
Time (sec)
0.40
0
GROUND ACC, g

0.20
-5
0.00

-10
Many ground motions now
-0.20
are available via the
-15
-0.40
0.00    1.00       2.00       3.00                         4.00          5.00          6.00                                                                                          0                     10        20      30         40       50                                           60                                     Internet.
TIME, SECONDS                                                                                                                                                                                   Time (sec)

Instructional Material Complementing FEMA 451, Design Examples                                            SDOF Dynamics 3 - 55                                                                                                   Instructional Material Complementing FEMA 451, Design Examples                                                                            SDOF Dynamics 3 - 56

Development of Effective Earthquake Force                                                                                                                                                                                           “Simplified” form of Equation of Motion:
&&
ut                                                                                                                                                                                                                                        mur (t ) + cu r (t ) + kur (t ) = −mu g (t )
&&         &                       &&
&&
ug              &&
ur
Divide through by m:
Ground Acceleration Response History
c          k
ur (t ) +
&&               ur (t ) + ur (t ) = −u g (t )
&                    &&
0.40
GROUND ACC, g

0.20
m          m
0.00

-0.20                                                                                                                                            Make substitutions:
c                                                                         k
= 2ξω                                                                     =ω2
-0.40
0.00          1.00          2.00          3.00      4.00     5.00       6.00
TIME, SECO S
ND

m                                                                         m
m[&&g ( t ) + ur ( t )] + c ur ( t ) + k ur ( t ) = 0
u           &&            &                                                                                                                                                                                              Simplified form:

mur ( t ) + c ur ( t ) + k ur ( t ) = − mug ( t )
&&           &                          &&                                                                                                                                                                                           u r (t ) + 2ξω ur (t ) + ω 2u r (t ) = −u g (t )
&&             &                        &&

Instructional Material Complementing FEMA 451, Design Examples                                            SDOF Dynamics 3 - 57                                                                                                   Instructional Material Complementing FEMA 451, Design Examples                                                                            SDOF Dynamics 3 - 58

For a given ground motion, the response                                                                                                                                                                                                  Response to Ground Motion (1940 El Centro)
history ur(t) is function of the structure’s                                                                                                                                                                                       0.4
Excitation applied to structure
with given ξ and ω
0.3

frequency ω and damping ratio ξ.
Ground Acceleration (g's)

0.2

0.1

0

Structural frequency                                                                                                                                                          -0.1
SOLVER
-0.2

-0.3

u r (t ) + 2ξω u r (t ) + ω 2 u r (t ) = −u&g (t )
0        10          20         30        40                                            50                                    60

&&             &                          &                                                                                                                                                                                                                          Time (sec)
6
Structural Displacement (in)

4                                                      Computed response
2
Change in ground motion
Damping ratio                                                                                                                                           or structural parameters ξ
0

and ω requires re-
-2
Ground motion acceleration history                                                                                                                                                                                                                                                                                    -4
calculation of structural                                                                                                                                                 Peak displacement
response                                                                                                              -6
0                                    10            20       30             40         50        60
Time (sec)
Instructional Material Complementing FEMA 451, Design Examples                                            SDOF Dynamics 3 - 59                                                                                                   Instructional Material Complementing FEMA 451, Design Examples                                                                            SDOF Dynamics 3 - 60

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                                                                                                Structural Dynamics 10
Computation of Response Spectrum for
The Elastic Displacement Response Spectrum                                                                                                                                                                                                                                                              El Centro Ground Motion
0.08
An elastic displacement response spectrum is a plot

Displacement, Inches
0.06
0.04                                                                         Computed response
of the peak computed relative displacement, ur, for an                                                                                                                                                                                                                                  0.02

elastic structure with a constant damping ξ, a varying
0.00
-0.02

fundamental frequency ω (or period T = 2π/ ω), responding
-0.04
-0.06
-0.08

to a given ground motion.                                                                                                                                                                                                                                                                       0     1     2       3             4                         5         6         7   8      9      10         11    12
Time, Seconds

5% damped response spectrum for structure                                                                                                                                                                                                                                         10.00
responding to 1940 El Centro ground motion                                                                                                                                                                                                                                                          Elastic response spectrum
16                                                                                                                                                                                                                                                   8.00

ξ = 0.05

Displacement, Inches
DISPLACEMENT, inches

12                                                                                                                                                                                                                                                   6.00

T = 0.10 sec                                                                             4.00
8

Umax= 0.0543 in.
4                                                                                                                                                                                                                                                    2.00

0.00
0
0.00              0.50          1.00              1.50         2.00
0                                        2                   4           6            8              10
Period, Seconds
PERIOD, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                                                                                 SDOF Dynamics 3 - 61                                                 Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 62

Computation of Response Spectrum
Computation of Response Spectrum
for El Centro Ground Motion
for El Centro Ground Motion
0.80
0.40

Displacement, Inches
0.60
Computed response
Displacement, Inches

0.30
0.20
Computed response                                                                                                  0.40
0.20
0.10
0.00
0.00
-0.20
-0.10
-0.40
-0.20
-0.60
-0.30
-0.80
-0.40
0     1     2       3             4                         5        6          7   8      9      10         11    12
0         1         2                            3                     4                             5            6           7       8           9        10          11         12
Time, Seconds
Time, Seconds

10.00                                                                                                                                                                                                 10.00

Elastic response spectrum                                                                                                                                                                       Elastic response spectrum
8.00                                                                                                                                                                                               8.00
Displacement, Inches
ξ = 0.05
Displacement, Inches

ξ = 0.05                                                                                                                                         6.00                                                                                                                                                                                               6.00

T = 0.20 sec                                                                                                                                     4.00
T = 0.30 sec                                                                             4.00

Umax = 0.254 in.                                                                                                                                                                                                                                           Umax = 0.622 in.                                                                         2.00
2.00

0.00                                                                                                                                                                                               0.00
0.00                    0.50                 1.00                  1.50                   2.00                                                                                                     0.00               0.50         1.00              1.50         2.00
Period, Seconds                                                                                                                                                                           Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                                                                                 SDOF Dynamics 3 - 63                                                 Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 64

Computation of Response Spectrum                                                                                                                                                                                              Computation of Response Spectrum
for El Centro Ground Motion
for El Centro Ground Motion
1.20
2.40
Displacement, Inches

0.90
Displacement, Inches

0.60
Computed response                                                                                         1.80
Computed response
1.20
0.30
0.60
0.00
0.00
-0.30
-0.60
-0.60
-1.20
-0.90
-1.80
-1.20
-2.40
0       1                        2             3                                     4              5           6           7        8          9         10          11         12
0     1     2       3             4                         5         6         7   8       9         10     11    12
Time, Seconds
Time, Seconds

10.00
10.00
Elastic response spectrum                                                                                                                                                                       Elastic response spectrum
8.00
8.00
Displacement, Inches

ξ = 0.05
Displacement, Inches

6.00                                                                                                         ξ = 0.05                                                                                 6.00

T = 0.40 sec                                                                                                                                  4.00
T = 0.50 sec                                                                             4.00

Umax = 0.956 in.                                                                                                                                                                                                                                           Umax = 2.02 in.
2.00
2.00

0.00
0.00
0.00                       0.50                 1.00              1.50                   2.00
0.00               0.50         1.00              1.50         2.00
Period, Seconds
Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                                                                                 SDOF Dynamics 3 - 65                                                 Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 66

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                                                                                                      Structural Dynamics 11
Computation of Response Spectrum                                                                                                                                                                               Complete 5% Damped Elastic Displacement
for El Centro Ground Motion                                                                                                                                                                                    Response Spectrum for El Centro
3.20
Ground Motion
Displacement, Inches

2.40                                                                                                                                                                                                                                 12.00
1.60                                                                                                                       Computed response
0.80
0.00
-0.80                                                                                                                                                                                                                                10.00
-1.60
-2.40
-3.20

Displacement, Inches
8.00
0                                                   1              2     3          4                       5       6          7     8       9         10     11           12
e,
Tim Seconds

10.00
6.00
Elastic response spectrum
8.00
Displacement, Inches

ξ = 0.05                                                                                                                                6.00
4.00

T = 0.60 sec                                                                                                                            4.00
2.00

Umax= -3.00 in.
2.00
0.00
0.0         0.5      1.0       1.5        2.0                        2.5          3.0          3.5         4.0
0.00
0.00                 0.50        1.00              1.50                2.00                                                                                                          Period, Seconds
Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                              SDOF Dynamics 3 - 67                                                                     Instructional Material Complementing FEMA 451, Design Examples                                      SDOF Dynamics 3 - 68

Development of Pseudovelocity                                                                                                                                                                        Development of Pseudoacceleration
Response Spectrum                                                                                                                                                                                      Response Spectrum
400.0
35.00
5% damping                                                                                                           350.0                                                                  5% damping
30.00
300.0
2
Pseudoacceleration, in/sec

25.00                                                                                                                                                                                                                        250.0
Pseudovelocity, in/sec

20.00                                                                                                                                                                                                                        200.0

15.00
150.0                                   PSA (T ) ≡ ω 2 D
100.0
10.00

PSV (T ) ≡ ω D
50.0
5.00
0.0
0.00                                                                                                                                                                                                                               0.0              1.0                     2.0                                3.0                     4.0
0.0                     1.0                                 2.0                3.0                      4.0                                                                                                                         Period, Seconds

Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                              SDOF Dynamics 3 - 69                                                                     Instructional Material Complementing FEMA 451, Design Examples                                      SDOF Dynamics 3 - 70

Note About the Pseudoacceleration Response Spectrum                                                                                                                                                                                                                                              PSA is TOTAL Acceleration!
The pseudoacceleration response spectrum represents the total                                                                                                                                                                           &&
ut
acceleration of the system, not the relative acceleration. It is nearly
identical to the true total acceleration response spectrum for lightly
&&
ug                                                          &&
ur
damped structures.
400.0
Ground Acceleration Response History
0.40

350.0                                                                   5% damping
GROUND ACC, g

0.20

300.0                                                                                                                                                                                                                                                      0.00
2
Pseudoacceleration, in/sec

-0.20
250.0

-0.40
200.0                                                                                                                                                                                                                                                          0.00         1.00         2.00       3.00         4.00   5.00      6.00
TIME, SECO S
ND
150.0

m[&&g ( t ) + ur ( t )] + c ur ( t ) + k ur ( t ) = 0
&&            &
100.0

Peak ground
u
50.0
acceleration
0.0
0.0                              1.0                  2.0               3.0                     4.0
mur ( t ) + c ur ( t ) + k ur ( t ) = − mug ( t )
&&           &                          &&
Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                              SDOF Dynamics 3 - 71                                                                     Instructional Material Complementing FEMA 451, Design Examples                                      SDOF Dynamics 3 - 72

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                                                                                      Structural Dynamics 12
Difference Between Pseudovelocity
Difference Between Pseudo-Acceleration
and Relative Velocity
and Total Acceleration                                                                                                                                                                                      (System with 5% Damping)
(System with 5% Damping)
40
350.00
35
300.00
Acceleration (in/sec )

30
2

Ve locity (in/se c)
250.00
25
200.00
20
150.00
15
100.00
10
50.00
5
0.00                                                                                                                                                      0
0.1                                    1                                       10
0.1                                          1                                       10
Period (sec)
Period (sec)
Total Acceleration                   Pseudo-Acceleration
Relative Velocity                 Pseudo-Velocity
Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 73                                                                          Instructional Material Complementing FEMA 451, Design Examples     SDOF Dynamics 3 - 74

Displacement Response Spectra                                                                                                                      Pseudoacceleration Response Spectra
for Different Damping Values                                                                                                                          for Different Damping Values
Damping                                                                                                                                                        Damping
25.00                                                                                                                                                             4.00
0%                                                                                                                                                               0%
5%                                                                                                                                                               5%
20.00
Displacement, Inches

Pseudoacceleration, g

10%                                                                                 3.00                                                                         10%
20%                                                                                                                                                              20%
15.00
2.00
10.00

1.00
5.00

0.00                                                                                                                                                             0.00
0.0           1.0             2.0              3.0            4.0        5.0                                                                                      0.0             1.0             2.0             3.0            4.0     5.0
Peak ground
Period, Seconds                                                                                                                                                     Period, Seconds
acceleration

Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 75                                                                          Instructional Material Complementing FEMA 451, Design Examples     SDOF Dynamics 3 - 76

Damping Is Effective in Reducing the                                                                                                                              Damping Is Effective in Reducing the
Response for (Almost) Any Given Period                                                                                                                              Response for Any Given Period of
of Vibration                                                                                                                                                      Vibration
4.00
Amplitude

2.00
0.00
• An earthquake record can be considered to be the                                                                                                                    -2.00
`

-4.00
combination of a large number of harmonic components.                                                                                                                                              0.0    6.0   12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0

• Any SDOF structure will be in near resonance with one                                                                                                                                                                                             Time (sec)
of these harmonic components.
• Example of an artificially generated wave to
• Damping is most effective at or near resonance.                                                                                                                   resemble a real time ground motion
accelerogram.
• Hence, a response spectrum will show reductions due to
damping at all period ranges (except T = 0).                                                                                                                    • Generated wave obtained by combining five
different harmonic signals, each having equal
amplitude of 1.0.

Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 77                                                                          Instructional Material Complementing FEMA 451, Design Examples     SDOF Dynamics 3 - 78

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                 Structural Dynamics 13
The Artificial Wave Is the Sum of Five Harmonics                                                                                                                                                                                                            The Artificial Wave Is the Sum of Five Harmonics
1                                                                                                                                                                                                                                     1
0.5                                                                                                                                                                                                                                   0.5
0                                                                                            `                                                                                                           T = 5.0 s                    0                                                                                            `
T = 2.0 s
-0.5                                                                                                                                                                                                                                  -0.5
-1                                                                                                                                                                                                                                    -1
0.0   6.0   12.0   18.0   24.0   30.0                            36.0          42.0                                   48.0        54.0     60.0    66.0       72.0     78.0       84.0     90.0
0.0   6.0                     12.0    18.0   24.0   30.0   36.0   42.0                         48.0       54.0     60.0     66.0         72.0     78.0    84.0     90.0

Amplitude
1
Amplitud

1
0.5
0.5
0                                                                                                                                                                                                         T = 4.0 s
-0.5
`
0                                                                                            `
T = 1.0 s
-1                                                                                                                                                                                                                                   -0.5
-1
e

0.0    6.0   12.0   18.0   24.0   30.0                       36.0           42.0                                       48.0     54.0        60.0    66.0       72.0     78.0       84.0      90.0
0.0   6.0                     12.0    18.0   24.0   30.0   36.0   42.0                        48.0        54.0     60.0     66.0     72.0         78.0    84.0     90.0

1                                                                                                                                                                                                                                                                                                                                                                                                           Summation
0.5                                                                                                                                                                                                                                   4.00
0                                                                                         `                                                                                                               T = 3.0 s                 2.00
-0.5                                                                                                                                                                                                                                   0.00                                                                                            `
-1                                                                                                                                                                                                                                  -2.00
0.0    6.0   12.0   18.0   24.0   30.0                       36.0           42.0                                       48.0     54.0        60.0    66.0       72.0     78.0       84.0      90.0                              -4.00
0.0   6.0                     12.0    18.0   24.0   30.0   36.0   42.0                        48.0        54.0     60.0   66.0       72.0        78.0     84.0    90.0

Time (sec)                                                                                                                                                                                                    Time (sec)

Instructional Material Complementing FEMA 451, Design Examples                                                                                                  SDOF Dynamics 3 - 79                                                                         Instructional Material Complementing FEMA 451, Design Examples                                                      SDOF Dynamics 3 - 80

Damping Reduces the Response                                                                                                                                                                                        Use of an Elastic Response Spectrum
at Each Resonant Frequency
Example Structure
12.00
14.00
K = 500 k/in
12.00                                                    0.0% Damping                                                                                                                                    10.00
5.0 % Damping
W = 2,000 k

Displacement, Inches
10.0% Damping
Dynamic Response Amplifier

10.00                                                    25.0 % Damping

M = 2000/386.4 = 5.18 k-sec2/in                                                                            8.00
Fourier amplitude

8.00

6.00
ω = (K/M)0.5 =9.82 rad/sec                                                                                 6.00

4.00                                                                                             T = 2π/ω = 0.64 sec                                                                                        4.00
2.00
5% critical damping
2.00
0.00
0.00          0.50      1.00         1.50       2.00          2.50       3.00
Frequency Ratio, β
0.00
0.0      0.5      1.0          1.5          2.0      2.5      3.0     3.5   4.0

Period, Seconds

Frequency (Hz)                                                                                                                                  At T = 0.64 sec, displacement = 3.03 in.
FFT curve for the combined wave

Instructional Material Complementing FEMA 451, Design Examples                                                                                                  SDOF Dynamics 3 - 81                                                                         Instructional Material Complementing FEMA 451, Design Examples                                                      SDOF Dynamics 3 - 82

Use of an Elastic Response Spectrum                                                                                                                                                                                                                        Response Spectrum, ADRS Space
400.0
Example Structure                                                                                                                                                                                                                                                                                   1.00
350.0
K = 500 k/in                                                                                                                                                                                                                                                                                                                                                                                                     Diagonal lines represent
Pseudoacceleration, in/sec 2

W = 2,000 k                                                                                            300.0                                                                                                                                                                                                                                                                                                     period values, T
0.80
Pseudoacceleration, g

M = 2000/386.4 = 5.18 k-sec2/in                                                                        250.0
T = 0.64s
ω = (K/M)0.5 =9.82 rad/sec                                                                             200.0
0.60
T = 2π/ω = 0.64 sec                                                                                    150.0
5% critical damping
100.0
0.40
50.0

0.0
0.0                                      0.5          1.0          1.5         2.0       2.5         3.0            3.5     4.0                                                      0.20
Period, Seconds

At T = 0.64 sec, pseudoacceleration = 301 in./sec2                                                                                                                                                                                                                                           0.00
0.00       2.00         4.00                               6.00                8.00            10.00                12.00
Base shear = M x PSA = 5.18(301) = 1559 kips                                                                                                                                                                                                                                                                                Displacement, inches

Instructional Material Complementing FEMA 451, Design Examples                                                                                                  SDOF Dynamics 3 - 83                                                                         Instructional Material Complementing FEMA 451, Design Examples                                                      SDOF Dynamics 3 - 84

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                                                                                                       Structural Dynamics 14
Four-Way Log Plot of Response Spectrum                                                                                                                                                                                                          Four-Way Log Plot of Response Spectrum
Line of increasing                                                                                                                                                                                                Line of constant
displacement                                                                                                                                                                                                      acceleration
100
100
D=10.0                                  1.0                       0.1                                                                                                                                                  PSA=1000                    10000                 100000
Line of constant                                                                                                                                                                                             Line of increasing
PSEUDOVELOCITY, in/sec

PSEUDOVELOCITY, in/sec
displacement                                                                                                                                                                                                 acceleration
10                                                                                                                                                                                                                                           10
1.0                                                0.01                                                                                                                                                           100                                         10000
PSA = PSV ω
PSV
D=
1

0.1                      .01                       0.001
ω                                                                     1

10                     100                   1000

0.1                                                                                                                                                                                                                                          0.1
0.1                                                    1                                 10                           100                    1000                                                                                            0.1                                        1                    10                     100                   1000

Circular Frequency ω, Radiand ω Second
Circular Frequency per                                                                                                                                                                                                                  Circular Frequency ω
Circular Frequency ω, Radiand per Second

Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 85                                                                                                       Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 86

Four-Way Log Plot of Response Spectrum
Four-Way Log Plot of Response Spectrum                                                                                                                                                                                                                                                                                                       Plotted vs Period
2
100                                                                                                                       ec                                                                                                                   100.00

.
DI

in
SP                                                                                                    /s
n
,i

t,
en
Ac
LA                                                                                               N
C                                                                                          IO

em
ce
10
0 EM                                                                                      AT 100

le
PSEUDOVELOCITY, in/sec

ac
EN                                                                       ER
PSEUDOVELOCITY, in/sec
00

ra

pl
EL

.0

10
T,

tio

is
10
in                                                       C

.0
n,

D
10                                       10                                                          AC 10

g
00                                                                                                                                 10.00

0

1.
1.

0
0                                  10
1.                                       0

0.
1
10
0.
1                                                                    1        10
0.
1.00

01
0.

0.
01
01
0.                                 1
1                                                     0.
1
00

1
0.
0.

00
00

0.1

0.
1

0.1                                     1                           10                         100                  1000
0.10
Circular Frequency Frequency ω Second
Circular ω, Radiand per                                                                                                                                                                                        0.01                   0.10                     1.00                   10.00
PERIOD, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 87                                                                                                       Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 88

Development of an Elastic                                                                                                                                                                                                       1940 El Centro, 0.35 g, N-S
Response Spectrum                                                                                                                                      100
.
in
t,
Ac

For a given earthquake,
en
ce

em

Problems with Current Spectrum:                                                                                                                                                                                                                small variations in structural
le

ac
ra

100.00                                                                                                                                                                                                                                                                                                                                                                                    frequency (period) can produce
pl
10
tio
.0
Ac

is
.

significantly different results.
.0
, in

10

n,
ce

D
Pseudo Velocity, In/Sec
t
le

en

g
ra

For a given earthquake,
m
t
io

ce
PSEUDOVELOCITY, in/sec

10
10
n

la
,g
.0

sp

.0
10

0
Di

1.

small variations in structural
1.

0

10.00
0% Damping
1.
0

frequency (period) can produce
1.

0

5% Damping
0.
10
1

10% Damping
0.

0.

significantly different results.
10
1
0.

20* Damping
1.00                                                                                                                                                                                                                                             1
0.

01
0.
01
01

0.

01

0.

It is for a single earthquake; other
0.

1
00

00
1

0.

1
0.

00

earthquakes will have different
00

0.10
0.
1

0.01                                                    0.10                               1.00                          10.00
PERIOD, Seconds                                                          Characteristics.                                                                                       0.1
0.01                                                                  0.1                1                                     10
Period, Seconds

Instructional Material Complementing FEMA 451, Design Examples                                          SDOF Dynamics 3 - 89                                                                                                       Instructional Material Complementing FEMA 451, Design Examples                      SDOF Dynamics 3 - 90

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                                                                                                                                                                             Structural Dynamics 15
5% Damped Spectra for Four California Earthquakes
Scaled to 0.40 g (PGA)                                                                                                                                      Smoothed Elastic Response Spectra
(Elastic DESIGN Response Spectra)
Different earthquakes
100.0
will have different spectra.
Pseuso Velocity, in/sec

10.0                                                                     El Centro
Loma Prieta
North Ridge
• Newmark-Hall spectrum
1.0
San Fernando
Average
• ASCE 7 spectrum

0.1
0.01                  0.10             1.00               10.00
Period, seconds

Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 91                                                 Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 92

Newmark-Hall Elastic Spectrum                                                                                                             Very Stiff Structure (T < 0.01 sec)

100
0% Dam ping
5% Dam ping
10% Dam ping

10
Displacem ent (in)

&
max u                                   Observations
v → max v
g

&&      &&     g

1
max u                     v→0        at short T
g
&&
max u   g
v → max v     g
at long T
v→0
&&
0.1
0.01                              0.1             1
Period (sec)          10           100                                                  Relative displacement                                                                           Zero
Total acceleration                                                                         Ground acceleration
Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 93                                                 Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 94

Very Flexible Structure (T > 10 sec)                                                                                                                  1940 El Centro, 0.35 g, N-S
100
Ac

.
in
ce

t,
en
le
ra

em
tio

ac
10
n,
.0

pl

.0
g
10

is
Pseudo Velocity, In/Sec

D

10
1.
0
1.

0

0% Damping
12.7 in/s                                   5% Damping
0.
10
1

10% Damping
0.

20* Damping
1                                                              4.25 in.
0.

01
01

0.

0.35g
Ground Maxima
0.

1

Relative displacement                                                                       Ground displacement
00

00
1

0.

Total acceleration                                                                      Zero                                                                    0.1
0.01                         0.1                1                               10
Period, Seconds
Instructional Material Complementing FEMA 451, Design Examples       SDOF Dynamics 3 - 95                                                 Instructional Material Complementing FEMA 451, Design Examples        SDOF Dynamics 3 - 96

FEMA 451B Topic 3 Handouts                                                                                                                                                                                                                                          Structural Dynamics 16
Newmark’s Spectrum Amplification Factors                                                                                    Newmark-Hall Elastic Spectrum
for Horizontal Elastic Response
1) Draw the lines
3                                    corresponding to max      && &
v ,v ,vg     g      g
Damping                       One Sigma (84.1%)                                                 Median (50%)
2                             4                     2) Draw line α A         &&
max v
% Critical                  aa    av    ad                                           aa           av    ad                                                                                    from Tb to Tc
g

.05                        5.10 3.84 3.04                                          3.68         2.59 2.01
3) Draw line α V        &
max v
1                          4.38 3.38 2.73                                          3.21         2.31 1.82                       5
6
from Tc to Td
g

2                          3.66 2.92 2.42                                          2.74         2.03 1.63                                    1
4) Draw line α D    max v
3                          3.24 2.64 2.24                                          2.46         1.86 1.52                                                                                     from Td to Te
g

5                          2.71 2.30 2.01                                          2.12         1.65 1.39
5) Draw connecting line
7                          2.36 2.08 1.85                                          1.89         1.51 1.29                                                                                     from Ta to Tb
10                         1.99 1.84 1.69                                          1.64         1.37 1.20
20                         1.26 1.37 1.38                                          1.17         1.08 1.01                  Ta       Tb           Tc           Td           Te Tf              6) Draw connecting line
from Te to Tf

Instructional Material Complementing FEMA 451, Design Examples           SDOF Dynamics 3 - 97                         Instructional Material Complementing FEMA 451, Design Examples    SDOF Dynamics 3 - 98

ASCE 7
Uses a Smoothed Design Acceleration Spectrum
The ASCE 7 Response Spectrum
“Short                                                                         SDS
1        Sa = 0.6     T + 0.4 SDS
period”                                                                        T0
acceleration                                                                                          is a uniform hazard spectrum based on
Spectral Response

SDS                                                                    2        Sa = SDS
Acceleration, Sa

2
probabilistic and deterministic seismic
“Long period”
SD1
acceleration
3        Sa =
T                          hazard analysis.
1                     3                                                       TS
SD1
4        Sa = L 2D1
T

4

TS                 T = 1.0                TL
Period, T
Note exceptions at larger periods

Instructional Material Complementing FEMA 451, Design Examples           SDOF Dynamics 3 - 99                         Instructional Material Complementing FEMA 451, Design Examples   SDOF Dynamics 3 - 100

FEMA 451B Topic 3 Handouts                                                                                                                                                                                       Structural Dynamics 17

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