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INDEX
Acceleration, 6
angular, 4
in building codes, 232, 233,
235, 238
constant acceleration method, 124, 129
d’Alembert’s principle, 9
dynamic nonlinear analysis, 187
earthquake ground motion, 18, 19
earthquake response in SDOF systems,
212, 213
elastic design response, 210
Holzer method, 159
linear acceleration method, 124, 127
linear elastic response, 203–206
measuring, 96
Newmark’s numerical method, 127
numerical integration methods,
124–127
and resonance, 87
seismic response analysis, 179
spectral analysis, 225
time-dependent force, 15–17, 24
undamped free vibration, 51
Amendments (building codes), 261
American Society of Civil Engineers
(ASCE), 251
Amplitude:
defined, 54
half-power, 92
Angular acceleration, 4
Angular momentum, 3, 4
Applied load, in static vs. dynamic
problems, 9
Applied Technology Council (ATC),
256
Applied Technology Council (ATC) 3–06
resource document for model
codes (1978), 256–261
1985 NEHRP 85, 256–258
1988 UBC, 258–159
1997 UBC, 259–261
Approximate analysis, 13
of earthquake ground motion response,
205
for short-duration impulse loads,
112–117
Arbitrary dynamic loading, 121–134
Duhamel integral, 121–122
Newmark’s numerical method,
127–134
numerical formulation of equation of
motion, 123
numerical integration methods,
124–127
step-by-step integration method,
123–124
Basic Structural Dynamics James C. Anderson and Farzad Naeim
Copyright © 2012 John Wiley & Sons, Inc.
265
266
INDEX
ASCE (American Society of Civil
Engineers), 251
ASCE, 7, 232–233, 261
Assumed deflection pattern method, 14
ATC (Applied Technology Council), 256
ATC 3–06, see Applied Technology
Council 3–06 resource document
for model codes (1978)
Attenuation relations, 211–212
Axial load effects, 10
multiple-degree-of-freedom systems,
162–164
single-degree-of-freedom systems,
42–44
Beating, 82
Bernoulli, Johann, 8
Betti’s theorem (law), 153–154
Blast-induced loads, 2
Block pulse, 101
Bozorgnia, Y., 212
Building code seismic provisions,
232–244
defining design criteria for buildings,
217
equivalent lateral force procedure,
234–237, 241–242
historical development of, 249–261
1978 ATC 3–06 resource
document for model codes,
256–261
2000–2009 International
Building Code, 261
overview, 250–256
modal response spectrum procedure,
234, 237–244
new generation of attenuation relations
in, 211–212
regional and local codes, 232
California:
California Building Code, 232, 261
California Strong Motion
Instrumentation Program, 246,
247
earthquakes in, 207, 209, 218–219,
224, 240, 250
Field Act, 250–251
model codes for California building
codes, 251–252
Riley Act, 250
California Building Code (CBC), 232, 261
California Strong Motion Instrumentation
Program (CSMIP), 246, 247
Campbell, K. W., 212
CBC (California Building Code), 232, 261
Changing coordinates, MDOF systems,
156–159
Circum-Pacific belt, 203
Close-coupled systems, 159
Clough, R., 166
Conservation laws, 8
Conservative forces, 8
Constant acceleration method, 124, 129
Constantinides, Margarita, 190
Constraints, 8
Continuous parameters, SDOF systems,
31–37
Critical damping, 61–62, 66
Crust (of Earth), 203
CSMIP (California Strong Motion
Instrumentation Program), 246,
247
D’Alembert, Jean le Rond, 8–9
D’Alembert’s principle, 8–9, 19–20
Damped dynamic systems, harmonic
loading response for, 84–90
Damped force vibration, 174–178
Damped free vibration, SDOF systems,
61–72
logarithmic decrement, 66–67
radical positive, 67–72
rotating vector form, 63–65
Damped rectangular pulse, 108
Damping:
critical, 61–62, 66
in discrete-parameter systems, 23
generalized, 24
and harmonic loading response, 91–95
overdamped systems, 67
viscous, 139–140
Damping coefficient, 10
Damping forces, 9, 23, 31, 139, 184
INDEX
Damping ratios, 177
and harmonic loading, 86, 92, 93
single-degree-of-freedom systems,
67–72
Dead load, 10
Deflected shape, 144–145
and earthquake motion, 216
frequency equation for, 143
fundamental period in transverse
direction, 55–58
selection of, 25, 60
static, 14, 22, 33, 56, 57
true, 58
Deflection:
assumed deflection pattern method, 14
reciprocal, Maxwell’s law of, 154
Deflection amplification factor (seismic
design), 234
Degrees of freedom, 8
and complexity of structural system,
10–11
defined, 8
reduction of, 13–14
Design spectra (earthquakes):
ASCE 7 definition of spectral
accelerations, 232–233
attenuation relations, 211–212
elastic design response spectrum,
208–215
Internet-based software for, 212
linear elastic response, 203–208
USGS Hazard Calculator,
213–215
Deterministic analysis, 1–2
Deterministic loads, periodic and
nonperiodic, 2
Discrete parameters, SDOF systems,
23–31
Discretized model, 13
Displacement(s), 13, 20–22
generalized coordinates in SDOF
systems, 22–46
axial load effect, 42–44
continuous parameters, 31–37
discrete parameters, 23–31
linear approximation, 44–46
transformation factors, 38–41
267
relative, 15, 22, 23, 173, 205, 215, 216
spectral analysis, 225
virtual:
defined, 8
in single-degree-of-freedom
systems, 20–22
Displacement function, 22
Displacement method, static equilibrium
equation in, 9
DLF (dynamic load factor), 79
Duhamel integral:
arbitrary dynamic loading, 121–122
seismic response, 203–206,
215–216
Dynamic (term), 1
Dynamic environment, 1–2
Dynamic equilibrium, 9–11. See also
Equation of motion
basic equation of, 9
under severe dynamic loading, 10
Dynamic force, effective, 16
Dynamic load factor (DLF), 79
Dynamic loads:
evaluating response to:
deterministic analysis, 1–2
nondeterministic analysis, 2
sources of, 2
types of, 2–3
Dynamic nonlinear analysis, MDOF
systems, 185–189
Earthquakes:
ground motion in SDOF systems,
18–19
and plate tectonics, 202–203
seismic accelerometers and
seismographs, 95–97
Earthquake engineering, 201. See also
Seismic response
Earthquake loads:
design spectra. See also Building code
seismic provisions
attenuation relations, 211–212
elastic design response spectrum,
208–215
Internet-based software for, 212
linear elastic response, 203–208
268
INDEX
Earthquake loads: (continued )
USGS Hazard Calculator,
213–215
duration of, 2
Effective dynamic force, 16
Eigenvalue equation, 141
Elastic design response spectrum,
208–215
Elastic properties, MDOF systems,
137–140
flexibility, 137–138
inertia, 139
stiffness, 138–139
viscous damping, 139–140
El Centro, California earthquake, 207,
209, 218–219, 240
Environmental loads, sources of, 2
Equation of motion, 9
arbitrary dynamic loading:
numerical formulation of,
123–124
numerical integration methods,
126
damped free vibration, 61
harmonic response:
damped dynamic systems, 84
undamped dynamic systems, 77
impulse loads, 101
damped rectangular pulse, 108
rectangular pulse, 104
seismic accelerometers and
seismographs, 95–96
seismic response, 179
damped oscillator subjected to
base acceleration, 203
linear elastic response, 206
SDOF systems, 215
undamped MDOF system with
time-dependent force, 185
undamped oscillator, 205
for single-degree-of-freedom systems,
19–22
d’Alembert’s principle,
19–20
earthquake response, 215
and gravitational forces, 17
and linear approximation, 44
in terms of generalized
coordinates and parameters, 24
virtual work (virtual
displacements), 20–22
undamped free vibration, 141, 155
Equilibrium equations:
time-dependent force, 15
undamped free vibration, 141
Equipment motions, loads from, 2
Equivalent lateral force procedure
(seismic response), 234–237,
241–242
Equivalent static pressures, 3
FEDEASLab toolbox (MATLAB),
190–200
Field Act (California), 250–251
Filippou, Filip C., 190
First (fundamental) mode, 142
Flexibility, in MDOF systems, 137–138
Flexure, 31, 36, 100
Force:
in continuous-parameter systems, 31
generalized, 24
Framing factor, 252
Free vibration, 51–72
damped, 61–72
logarithmic decrement, 66–67
radical positive, 67–72
rotating vector form, 63–65
defined, 51
multiple-degree-of-freedom systems,
141–153
single-degree-of-freedom systems,
61–72
undamped, 51–60, 141–149, 155
Rayleigh’s method, 58–60
rotating vector form, 53–58
selection of deflected shape, 60
single-degree-of-freedom
systems, 51–60
Free-vibration decay, 91
Frequency equation, 142
Fundamental (first) mode, 142
Gauss Reduction method, MDOF
systems, 189–190
INDEX
Generalized coordinate(s):
defined, 22
for multiple-degree-of-freedom
systems, 156
for single-degree-of-freedom systems,
22–46
axial load effect, 42–44
continuous parameters, 31–37
discrete parameters, 23–31
linear approximation, 44–46
transformation factors, 38–41
Generalized parameters, 24
Geometric nonlinearity, 10
Gravitational forces, SDOF systems, 17
Gravity loads, 10
Half-power amplitude, 92
Half-power method, 92–93
Harmonic loading, 77–97
damped dynamic systems, 84–90
evaluation of damping, 91–95
seismic accelerometers and
seismographs, 95–97
tripartite logarithmic plot, 91, 92
undamped dynamic systems, 77–84
Harmonic motion:
at constant circular frequency, 63
simple, 52–53
Holzer method for shear buildings,
159–162
Housner, G. W., 209
Housner’s earthquake design spectrum,
208–209
IBC, see International Building Code
ICBO (International Conference of
Building Officials), 250
Imperial Valley, California earthquake,
207, 218–219, 224
Importance factors (seismic design), 233
Impulse loads, 101–117
damped rectangular pulse, 108
rectangular pulse, 104–107
short-duration, approximate analysis
for, 112–117
triangular pulse, 109–112
Inertia, 139
269
Inertia forces, 10, 25, 31
causes, 23
components of, 139
d’Alembert’s principle, 9
damped forced vibration, 174
spectral analysis, 225
in systems with distributed mass, 216,
217
time-dependent, 15, 19
undamped free vibration, 155
International Building Code (IBC):
2000–2009 seismic provisions, 261
2009 IBC, 232
ASCE 7 provisions adopted by,
232–233, 261
NEHRP provisions adopted by, 261
and USGS Hazard Calculator,
213–215
International Conference of Building
Officials (ICBO), 250
Inverse transformation, MDOF systems,
156–159
Japan, seismic design law in, 250, 253
Kanto, Japan earthquake, 250
Kinetic energy (KE), 6, 58, 59
Linear acceleration method, 124
Linear approximation:
multiple-degree-of-freedom systems,
162–164
single-degree-of-freedom systems,
44–46
Linear elastic response spectra (seismic
response), 203–208
Linear momentum, Newton’s second law
of, 3
Linear variation, 126–127
Live load, 10
Load-mass factor, 39–41
Logarithmic decrement:
damped free vibration, 66–67
defined, 66
Long Beach, California earthquake, 250
Long-duration nonperiodic loading, 2
Long-duration nonperiodic loads, 2
270
INDEX
Los Angeles Building Code, 232, 251,
261
Lumped parameter method, 14
Mass, 3–10, 123
and arbitrary dynamic loading, 123, 125
and building codes, 250
in continuous-parameter systems, 31
and d’Alembert’s principle, 19
and damped force vibration, 174–176
in discrete-parameter systems, 23
distributed, 216
and earthquake ground motion, 18, 19
and elastic properties, 139, 140
generalized, 24, 28, 32, 36, 111, 165,
166
and gravitational forces, 17
lumped, 14, 19, 23, 139–140, 159, 217
and orthogonal mode shapes, 156, 157
properties of common shapes, 7
and seismic response, 179, 216–217,
221
and time-dependent force, 15
and transformation factors, 38–41
in undamped dynamic systems, 51
in undamped time-dependent force
analysis, 165–167
and undamped vibration, 51
weight vs., 6
Mass moment of inertia, 4–7
Mass participation factors (building
codes), 237, 241, 243–244
Material nonlinearity, 10
MATLAB applications:
arbitrary dynamic loading, 129–134
FEDEASLab toolbox, 190–200
harmonic loading:
damped dynamic system, 89–90
evaluation of damping, 93–95
in undamped dynamic systems,
80–81, 83–84
impulse loads:
rectangular pulse, 106–107
short-duration, 114–117
multiple-degree-of-freedom systems,
190–200
free vibration, 151–153
inverse transformation, 158–159
seismic response, 226–232,
237–240, 243–244
undamped free vibration,
145–149
undamped time-dependent force
analysis, 169–170, 173–174
Plot Tool, 70–72
seismic response, 226–232, 237–240,
243–244
single-degree-of-freedom systems:
axial load effect, 43–44
with continuous parameters,
33–35, 37
damping ratios, 68–72
rotating vector form, 63–65
undamped free vibration, 54–55
Symbolic Math Toolbox, 33–35, 37,
43–44, 151–153, 169–170
Maxwell’s law of reciprocal deflections,
154
MDOF systems, see
Multiple-degree-of-freedom
systems
Modal coordinates, see Normal
coordinates
Modal equations, MDOF systems:
of damped force vibration, 174–178
for seismic response analysis, 179–180
for undamped time-dependent force
analysis, 165–174
Modal response spectrum procedure
(seismic response), 234,
237–244
Modal superposition method, 166
Model codes:
1978 ATC 3–06 resource document for
model codes, 256–261
for California building codes, 251–252
International Building Code, 232, 261
Moment of inertia:
defined, 4
mass, 4–7
Moment of momentum, 4
Momentum:
angular, 3, 4
defined, 3
INDEX
linear, 3
Newton’s second law of, 3
Motion. See also Equation of motion
earthquake ground motion, see
Earthquakes
equipment, 2
Newton’s laws of, 3–8
properties of, 8
rotational, 4. See also Angular
momentum
Multiple-degree-of-freedom (MDOF)
systems, 137–180
axial load effects (linear
approximation), 162–164
Betti’s theorem (law), 153–154
changing coordinates (inverse
transformation), 156–159
elastic properties, 137–140
flexibility, 137–138
inertia, 139
stiffness, 138–139
viscous damping,
139–140
free vibration, 149–153
Holzer method for shear buildings,
159–162
modal equations:
of damped force vibration,
174–178
for seismic response analysis,
179–180
for undamped time-dependent
force analysis, 165–174
nonlinear response of, 183–200
dynamic nonlinear analysis,
185–189
Gauss Reduction method,
189–190
MATLAB applications,
190–200
static nonlinear analysis,
184–188
orthogonality properties of mode
shapes, 155–156
seismic response, 219–232
modal combinations for spectral
analyses, 222–232
271
time history modal analysis,
221–222
undamped free vibration, 141–149
Multiple degrees of freedom, 10–11
National Earthquake Hazard Reduction
Program (NEHRP), 256
IBC provisions from, 261
NEHRP 85, 256–258
Natural frequency:
effect of damping on, 62–63, 68
and harmonic loading, 93
Holzer method, 160
seismic measurement, 96
in undamped dynamic systems, 77–79,
81, 83
NEHRP, see National Earthquake Hazard
Reduction Program
NEHRP 85(1985), 256–258
New generation of attenuation (NGA)
relations, 211–212
Newmark, N. M., 127
Newmark-Hall earthquake design
spectrum, 209–210
Newmark’s numerical method, arbitrary
dynamic loading, 127–134
Newton, Sir Isaac, 3
Newton’s laws of motion, 3–8
first law, 3
second law, 3, 4, 7
third law, 3
Newton’s method, 160
NGA (new generation of attenuation)
relations, 211–212
Nonconservative forces, 8
Nondeterministic analysis, 2
Nonlinearity:
geometric, 10
material, 10
most common source of, 183
Nonlinear response, MDOF systems,
183–200
dynamic nonlinear analysis, 185–189
Gauss Reduction method, 189–190
MATLAB applications, 190–200
static nonlinear analysis, 184–188
Nonperiodic loads, 2
272
INDEX
Normal (modal) coordinates:
defined, 156
inverse transformation, 156–159
seismic response analysis, 219–220
Normal mode of vibration, 155–156
Northridge earthquake, 245–246, 248
Numerical integration methods:
arbitrary dynamic loading, 124–134
earthquake response of SDOF systems,
215
Newmark’s numerical method,
127–134
Occupancy categories, 233
Occupancy load, 10
Olympia, Washington earthquake, 209
Orthogonality properties of mode shapes,
MDOF systems, 155–156
Overdamped systems, 67
Parameters:
generalized, 24
single-degree-of-freedom systems:
continuous, 31–37
discrete, 23–31
P- effect, 10
PE (potential energy), 6, 58–59
Penzien, J., 166
Periodic loads, 2
Period of vibration, 253
fundamental, calculating, 235
and Rayleigh’s method, 59
using transformation factors, 39
Phase angle (term), 54
Plates, 202, 203
Plate tectonics, 202–203
Potential energy (PE), 6, 58–59
Principia (Sir Isaac Newton), 3
Principle of virtual work, 8
Properties of motion, 8
Pushover analysis:
defined, 185
for multiple-degree-of-freedom
systems, 184–188
Radical positive (damped free vibration),
67–72
Random dynamic loading, 2
Rayleigh, Lord, 14, 175
Rayleigh damping, MDOF systems,
175–176
Rayleigh’s method:
damped forced vibration, 175–176,
185–187
multiple-degree-of-freedom systems,
175–176, 185–187
single-degree-of-freedom systems:
for discrete parameters,
22–31
undamped free vibration,
58–60
Reciprocal deflections, Maxwell’s law of,
154
Rectangular pulse:
damped, 108
ideal, 104–107
Relative displacement, 15, 22, 23, 173,
205, 215, 216
Relative velocity, 15, 22, 205
Resonance, 79–80, 86–88
and damping, 92
defined, 79
Resonate amplification, 92
Response modification factor (seismic
design), 234
Response ratio, 79–80
Restoring force, 22, 23, 32, 123
damped force vibration, 174
static nonlinear analysis, 184
time-dependent force, 15
Riley Act (California), 250
Rotating vector form, SDOF systems:
damped free vibration, 63–65
undamped free vibration, 53–58
Rotational inertia, 5
Rotational motion, 4
San Francisco building code, 252
San Francisco earthquake, 250
SDOF systems, see
Single-degree-of-freedom
systems
SEAOC (Structural Engineers Association
of California), 252, 253
INDEX
SEAONC (Structural Engineers
Association of Northern
California), 251
Seismic accelerometers, 95, 96
Seismic design categories, 233, 234
Seismic response, 201–244
and building codes, 232–244
equivalent lateral force
procedure, 234–237, 241–242
modal response spectrum
procedure, 234, 237–244
elastic design response spectrum,
208–215
linear elastic response spectra, 203–208
multiple-degree-of-freedom systems,
219–232
modal combinations for spectral
analyses, 222–232
modal equations for, 179–180
time history modal analysis,
221–222
single-degree-of-freedom systems,
215–219
Seismic zones, 253, 254
Seismographs, 95–97
Self-weight, 10
Severe dynamic loading, nonlinear
behavior under, 10
Shanghai World Financial Center, 185,
187, 188
Shape function, 14, 22, 24–25, 30, 58
Shear, 31
base, 185, 217–218, 234–242, 244,
251–254, 256, 258–260
impulse loads, 103
lateral, 251
seismic design, 217–218, 221, 225,
226, 234–242, 244
spectral analysis, 225
static nonlinear analysis, 185, 186
story, 236, 242, 244, 246
Shear buildings:
Holzer method for, 159–162
Newton’s method, 160
undamped free vibration, 143
Shear building approximation, 140, 142
Shock spectra, 110–112
273
Short-duration loads:
impulse, approximate analysis for,
112–117
nonperiodic, 2
Simple harmonic loading, 2
Simple harmonic motion, 52–53
Single degree of freedom, 10
Single-degree-of-freedom (SDOF)
systems, 13–46
damped free vibration, 61–72
and earthquake ground motion, 18–19
formulation of equation of motion,
19–22
d’Alembert’s principle, 19–20
virtual work (virtual
displacements), 20–22
free-vibration response of, 51–72
damped free vibration, 61–72
undamped free vibration, 51–60
generalized coordinates for, 22–46
axial load effect, 42–44
continuous parameters, 31–37
discrete parameters, 23–31
linear approximation, 44–46
transformation factors, 38–41
gravitational forces in, 17
reduction of degrees of freedom, 13–14
seismic response, 215–219
with time-dependent force, 15–17
undamped free vibration, 51–60
Spatial shape function, 22
Spectral analyses, seismic response,
222–232
Static deflected shape method, 14
Static equilibrium:
in displacement method, 9
equation of, 9
Static nonlinear (pushover) analysis,
MDOF systems, 184–188
Steady state, 82
Step-by-step integration method, 123–124
Stiffness:
axial load effects, 162–164
in continuous-parameter systems, 31
as function of time, 123
Gauss reduction, 190
generalized, 22, 24, 33, 36
274
INDEX
Stiffness: (continued )
geometric, 44–45, 183, 185, 194, 196,
198
global, 190
Holzer method for shear buildings, 159
lateral, 29, 45, 162, 183
lateral force-resisting elements, 26
mode shapes, 156
multiple-degree-of-freedom systems,
137–139
static nonlinear analysis, 184, 186
story, 56–58, 140
string, 44
with time-dependent force, 15
undamped free vibration, 142, 146
undamped time-dependent force
analysis, 165–167
for very large structural systems, 190
Stiffness proportional damping,
175, 176
Story stiffness, 56–58, 140
String stiffness, 44
Structural elements, in discrete-parameter
systems, 23
Structural Engineers Association of
California (SEAOC), 252, 253
Structural Engineers Association of
Northern California (SEAONC),
251
Superposition, modal, 166
Superposition principle, in elastic
systems, 10
Symbolic Math Toolbox (MATLAB),
33–35, 37, 43–44, 151–153,
169–170
System overstrength factor (seismic
design), 234
Taft, California earthquake, 209
Time dependence:
earthquake ground motion, 19
equation of dynamic equilibrium, 123
in static vs. dynamic problems, 9
Time-dependent force:
generalized parameters:
continuous parameters, 31
discrete parameters, 23, 24
multiple-degree-of-freedom systems:
damped, 185
undamped, 165–174
single-degree-of-freedom systems,
15–17
Time history modal analysis (seismic
response), 221–222
Torque, 4
Traite de Dynamique (Jean le Rond
d’Alembert), 8–9
Transformation factors, SDOF systems,
38–41
Transient response, 82
Transitional mass, 7
Triangular pulse, 109–112
Tripartite logarithmic plot:
harmonic loading response, 91, 92
Newmark-Hall earthquake design
spectra, 210–211
seismic response, 206–208
True vibration shape, 25
UBC, see Uniform Building Code
Undamped (term), 51
Undamped dynamic systems:
Duhamel integral, 121–122
harmonic loading response, 77–84
time-dependent force analysis for
undamped MDOF systems,
165–174
Undamped free vibration:
multiple-degree-of-freedom systems,
141–149
single-degree-of-freedom systems,
51–60, 67
Rayleigh’s method, 58–60
rotating vector form, 53–58
selection of deflected shape, 60
Uniform Building Code (UBC):
1961 provisions, 253
1967 provisions, 254
1976 provisions, 254–256
1985 provisions, 256
1988 provisions, 258–259
1997 provisions, 259–261
enactment of seismic provisions,
250
INDEX
US Geological Survey (USGS), 212
USGS Hazard Calculator,
213–215
Vehicle loads, 2
Velocity, relative, 15, 22, 205
Virtual displacement:
defined, 8
single-degree-of-freedom systems,
20–23
Virtual work:
principle of, 8
275
single-degree-of-freedom systems,
20–23
Viscous damping:
harmonic loading, 84
multiple-degree-of-freedom systems,
139–140
Weight, mass vs., 6
Wind loads, 2–3
Work:
concept of, 6–7
virtual, 8, 20–22