DYNAMICS OF STRUCTURES Third Edition
3.0 Yielding phase
2.0
Displacement v, in
Elastoplastic response
Static displacement p = ¾ k
1.0 vi = inelastic displacement
ve = elastic limit
0 Elastic response
- 1.0 0
0.2
0.4
0.6
0.8
1.0
Time, sec FIGURE E7-4 Comparison of elastoplastic with elastic response (frame of Fig. E7-3).
fs
fs
8k
8k 1 in.
8k
v
1.5 in.
v
1.5 in. 8k
fs = 12 [ 2 v 3
( 1.5 (a) FIGURE P7-1
(b)
1 3
3
( 23v ) ] v
1.5)
CHAPTER
GENERALIZED SINGLEDEGREEOF-FREEDOM SYSTEMS
a 2 L 2
2
L 12
j=m
a 2 m
b 2
j=m
a 2+ b 2 12
)
m = ab
m = mL L 2
(
b 2
mass m= length mass
= area
m
2b 3
j=m
(
2
a +b 18
2
m= b 3 a 3
)
b 2
m
j=m
(
a 2+ b 2 16
m= ab 2
b 2
)
ab 4
Ellipse a 2
2a 3
a 2
FIGURE 8-1 Rigid-body mass and centroidal mass moment of inertia for uniform rod and uniform plates of unit thickness.
x
p(x, t) = p ax f (t)
A B
a
c1
m 2a
D
Hinge m2 , j2 E
k1 a
H G k2
c2 a
Weightless, rigid bar EH
a
FIGURE E8-1 Example of a rigid-body-assemblage SDOF system.
a
N
1
1
2
2
1
2
1
2
p1(t) = 8 pa f (t)
8a 3
A
B
Mj B
fD (t) 1
E
D
C fI (t) 1
F Mj
1
D fS (t) 1
E fD (t) 2
FIGURE E8-2 SDOF displacements and resultant forces.
F fI (t) 2
G
H
2
G fS (t) 2
Z(t)
e1
E
Z
e1
E
E
Z E
Z(t)
A
H
H e
4a
3a
FIGURE E8-3 Displacement components in the direction of axial force.
N
fS (t)
mass
= area (uniform)
a
k fI (t) 1
I (t)
b
fI (t)
b
2
2
Z(t) a 2
FIGURE E8-4 SDOF plate with dynamic forces.
p(t)
1
2
1
2
N Z(t)
e(t)
v t (x,t)
Fixed reference axis
v(x,t)
pe ff (x,t) =
m(x) v¨g (t)
L
m(x) EI(x)
x
(b)
vg (t) (a)
FIGURE 8-2 Flexure structure treated as a SDOF system.
2 2
(a)
v(x,t) = (x) Z(t)
v1
v2
v3 x
v4
Z (t)
L m(x)
m1, j1
m4 , j4
(b) x1
(c)
(d )
c(x)
k (x)
c2
a1(x)
k1
c3
EI (x)
k2
q (x) (e)
N
p(x,t)
p1 (t)
p3 (t)
(f) FIGURE 8-3 Properties of generalized SDOF system: (a) assumed shape; (b) mass properties; (c) damping properties; (d ) elastic properties; (e) applied axial loading; ( f ) applied lateral loading.
y
Z (t) b
w (x,y,t)
x a FIGURE 8-4 Simply supported two-dimensional slab treated as a SDOF system.
v k
m
(a) v v0 t
2
. v
2
(b)
v0 t
(c)
FIGURE 8-5 Free vibration of undamped SDOF structure: (a) SDOF structure; (b) displacement; (c) velocity.
v(x,t) =
x
EI(x) m(x) L
(x) Z(t)
FIGURE 8-6 Vibration of a nonuniform beam.
Approximate inertial loading: ¾ p (x) = m (x) ¾ y(x) [where ¾ y(x) is assumed shape]
Computer deflected shape vd (x) »
¾
y (x)
FIGURE 8-7 Deflected shape resulting from inertial load of assumed shape.
p(x) = m(x) g
p(x) = m(x) g
vd (x) vd (x)
p(x) = m(x) g
vd (x)
p(x) = m(x) g (c)
(a)
(b)
FIGURE 8-8 Assumed shapes resulting from dead loads.
Weight = W
L 2
L 2
Uniform beam EI = stiffness m = mass length Assumed shape x
v(x) = Z 0
3x 2L x 3 3 2L
p
Z0 =
pL3 3EI
FIGURE E8-5 Rayleigh method analysis of beam vibration frequency.
m 1
2
3
kips sec 2 in k
1.5
v1(0) = 1.0 kips in v2(0) = 1.0
1,200
2.0
v3(0) = 1.0 1,800
(a) Inertial loads =
(b) 2
mi vi(0)
1.0
Computed deflections
2
(Shear force = 1.0 1.5
)
va =
600
2
(Shear = 2.5 2.0
2
2
)
vb =
2
v2(1) =
2.5 1,200
v3(1) =
2
(Shear = 4.5
v1(1) =
2
2
)
vc =
2
22.5 3,600
16.5 3,600
9.0 3,600
2
2
2
= Z 0(1)(1.0)
2
= Z 0(1)(0.733)
2
= Z 0(1)(0.40)
2
4.5 1,800
(c) FIGURE E8-6 Frame for Rayleigh method frequency analysis: (a) mass and stiffness values; (b) initial assumed shape; (c) deflections resulting from initial inertial forces.
2L p (t)
L
L
{
load length
Z (t) Rigid uniform bar Total mass = m
Pulley: Total mass = m (uniform over area) Inextensible massless cable
c
k FIGURE P8-2
p(t)
Rigid massless bar Rigid uniform bar (Total mass = m)
k
k Z(t) L
c L 2
L
L 2
FIGURE P8-3
x
Z (t)
v(x,t) =
(x) Z(t)
p=
L
load length
Uniform column m=
mass length
EI = flexural rigidity
FIGURE P8-4
x 2
8 ft
(x) = 1
200 ft
1
cos
x
2L
Concrete stack: density = 150 lb/ft3 E = 3 10 6 lb/in2 wall thickness = 8 in
0 18 ft
FIGURE P8-5
{
m1
m = mass length EI
x
Assumed shape L 2
v(x)
L 2
FIGURE P8-6
v1(0) = 1
m1 k1
v2(0) = 2 3
v3(0) = 1 3
m2 k2 m3 k3
kips sec2/in kips/in kips sec2/in kips/in kips sec2/in kips/in FIGURE P8-8
CHAPTER
FORMULATION OF THE MDOF EQUATIONS OF MOTION
p(x,t)
{
1 m(x) EI(x) v1 (t)
i
2
v 2 (t)
N
vi (t)
vN (t)
FIGURE 9-1 Discretization of a general beam-type structure.
CHAPTER
EVALUATION OF STRUCTURALPROPERTY MATRICES
~ v 1 = f 11
~ f 21 p1 = 1
~ f 12
~ fN 1
~ f i1
~ vi = fi 2 ~ f 22 p2 = 1
~ fN 2 FIGURE 10-1 Definition of flexibility influence coefficients.
p 1 = k 11 p 2 = k 21
pi 1 = k i 1
pN 1 = k N 1
pi = k i 2
pN = k N 2
v1 =1
p 2 = k 22 p 1 = k 12 v2 =1
FIGURE 10-2 Definition of stiffness influence coefficients.
Load system a: p1 a
Load system b: p2 a
p1 b
p3 a
p2 b
p3 b
Deflections b:
Deflections a:
v3 b v1 a
v2 a
v3 a
FIGURE 10-3 Two independent load systems and resulting deflections.
v1 b
v2 b
v(x)
EI(x)
a
b L x
va
v1 = 1
a
1 (x)
v3 = 1 3 (x)
FIGURE 10-4 Beam deflections due to unit nodal displacements at left end.
v(x) = va v3
a
=1
v1
1 (x)
v1
3 (x)
pa = k 13
FIGURE 10-5 Beam subjected to real rotation and virtual translation of node.
v2
v3 4EI v1
L
EI
EI 2L (a) k 21 = 2EI (3L) L3
k 31 = 2EI (3L) L3 2EI
k 11 = v1 = 1
L3
(6)(2)
(b) k 22 = =
2EI L3 2EI L3
(2L 2 ) +
2(4EI ) (2L) 3
(2)(2L) 2
k 32 =
(6L 2 )
2(4EI) (2L) 3
k 12 =
(2L) 2 =
2EI L3
2EI L3
(2L 2 )
(3L)
v2 = 1
(c)
FIGURE E10-1 Analysis of frame stiffness coefficients: (a) frame properties and degrees of freedom; (b) forces due to displacement v1 = 1; (c) forces due to rotation v2 = 1.
0
1
2
i
m0 a
m1 a
m2 c
mi k
m1 b
m2 b
mi l
m1
m2
mi
N
mN
FIGURE 10-6 Lumping of mass at beam nodes.
v(x)
m(x) b
a L v(x) = v1 v¨3
1 (x)
v1
va
¨ =1 a
Inertial force fI (x) pa = m 13
FIGURE 10-7 Node subjected to real angular acceleration and virtual translation.
v2
v3 1.5 m
L
v1
m
1.5 mL
m11 = 4 mL
1.5 mL
0.5 mL
0.5 mL
m 2L
0.5 mL
0.5 mL
(a) m 21 =
(b)
mL mL (22L) = (11L) 420 210
m 31 =
mL (11L) 210
(Axial motion of girder)
mL (156)(2) + (1.5 m)(2L) 420 mL = (786) 210
m 11 = v¨ 1 = 1
(c) m 22 = =
(1.5 m)(2L) mL (4L2) + (4)(2L)2 420 420 mL (26L2) 210
m 32 =
(1.5 m)(2L) mL ( 3)(2L2)2 = ( 18L2) 420 210
m 12 =
mL mL (22L) = (11L) 210 420
v¨ 2 = 1 (d) FIGURE E10-2 Analysis of lumped- and consistent-mass matrices: (a) uniform mass in members; (b) lumping of mass at member ends; (c) forces due to acceleration v¨1 = 1 (consistent); (d ) forces due to acceleration v¨2 = 1 (consistent).
( , )
=
1
1
( )=
1(
)
1
3.0 Yielding phase
2.0
Displacement v, in
Elastoplastic response
Static displacement p = ¾ k
1.0 vi = inelastic displacement
ve = elastic limit
0 Elastic response
- 1.0 0
0.2
0.4
0.6
0.8
1.0
Time, sec FIGURE E7-4 Comparison of elastoplastic with elastic response (frame of Fig. E7-3).
fs
fs
8k
8k 1 in.
8k
v
1.5 in.
v
1.5 in. 8k
fs = 12 [ 2 v 3
( 1.5 (a) FIGURE P7-1
(b)
1 3
3
( 23v ) ] v
1.5)
CHAPTER
GENERALIZED SINGLEDEGREEOF-FREEDOM SYSTEMS
a 2 L 2
2
L 12
j=m
a 2 m
b 2
j=m
a 2+ b 2 12
)
m = ab
m = mL L 2
(
b 2
mass m= length mass
= area
m
2b 3
j=m
(
2
a +b 18
2
m= b 3 a 3
)
b 2
m
j=m
(
a 2+ b 2 16
m= ab 2
b 2
)
ab 4
Ellipse a 2
2a 3
a 2
FIGURE 8-1 Rigid-body mass and centroidal mass moment of inertia for uniform rod and uniform plates of unit thickness.
x
p(x, t) = p ax f (t)
A B
a
c1
m 2a
D
Hinge m2 , j2 E
k1 a
H G k2
c2 a
Weightless, rigid bar EH
a
FIGURE E8-1 Example of a rigid-body-assemblage SDOF system.
a
N
1
1
2
2
1
2
1
2
p1(t) = 8 pa f (t)
8a 3
A
B
Mj B
fD (t) 1
E
D
C fI (t) 1
F Mj
1
D fS (t) 1
E fD (t) 2
FIGURE E8-2 SDOF displacements and resultant forces.
F fI (t) 2
G
H
2
G fS (t) 2
Z(t)
e1
E
Z
e1
E
E
Z E
Z(t)
A
H
H e
4a
3a
FIGURE E8-3 Displacement components in the direction of axial force.
N
fS (t)
mass
= area (uniform)
a
k fI (t) 1
I (t)
b
fI (t)
b
2
2
Z(t) a 2
FIGURE E8-4 SDOF plate with dynamic forces.
p(t)
1
2
1
2
N Z(t)
e(t)
v t (x,t)
Fixed reference axis
v(x,t)
pe ff (x,t) =
m(x) v¨g (t)
L
m(x) EI(x)
x
(b)
vg (t) (a)
FIGURE 8-2 Flexure structure treated as a SDOF system.
2 2
(a)
v(x,t) = (x) Z(t)
v1
v2
v3 x
v4
Z (t)
L m(x)
m1, j1
m4 , j4
(b) x1
(c)
(d )
c(x)
k (x)
c2
a1(x)
k1
c3
EI (x)
k2
q (x) (e)
N
p(x,t)
p1 (t)
p3 (t)
(f) FIGURE 8-3 Properties of generalized SDOF system: (a) assumed shape; (b) mass properties; (c) damping properties; (d ) elastic properties; (e) applied axial loading; ( f ) applied lateral loading.
y
Z (t) b
w (x,y,t)
x a FIGURE 8-4 Simply supported two-dimensional slab treated as a SDOF system.
v k
m
(a) v v0 t
2
. v
2
(b)
v0 t
(c)
FIGURE 8-5 Free vibration of undamped SDOF structure: (a) SDOF structure; (b) displacement; (c) velocity.
v(x,t) =
x
EI(x) m(x) L
(x) Z(t)
FIGURE 8-6 Vibration of a nonuniform beam.
Approximate inertial loading: ¾ p (x) = m (x) ¾ y(x) [where ¾ y(x) is assumed shape]
Computer deflected shape vd (x) »
¾
y (x)
FIGURE 8-7 Deflected shape resulting from inertial load of assumed shape.
p(x) = m(x) g
p(x) = m(x) g
vd (x) vd (x)
p(x) = m(x) g
vd (x)
p(x) = m(x) g (c)
(a)
(b)
FIGURE 8-8 Assumed shapes resulting from dead loads.
Weight = W
L 2
L 2
Uniform beam EI = stiffness m = mass length Assumed shape x
v(x) = Z 0
3x 2L x 3 3 2L
p
Z0 =
pL3 3EI
FIGURE E8-5 Rayleigh method analysis of beam vibration frequency.
m 1
2
3
kips sec 2 in k
1.5
v1(0) = 1.0 kips in v2(0) = 1.0
1,200
2.0
v3(0) = 1.0 1,800
(a) Inertial loads =
(b) 2
mi vi(0)
1.0
Computed deflections
2
(Shear force = 1.0 1.5
)
va =
600
2
(Shear = 2.5 2.0
2
2
)
vb =
2
v2(1) =
2.5 1,200
v3(1) =
2
(Shear = 4.5
v1(1) =
2
2
)
vc =
2
22.5 3,600
16.5 3,600
9.0 3,600
2
2
2
= Z 0(1)(1.0)
2
= Z 0(1)(0.733)
2
= Z 0(1)(0.40)
2
4.5 1,800
(c) FIGURE E8-6 Frame for Rayleigh method frequency analysis: (a) mass and stiffness values; (b) initial assumed shape; (c) deflections resulting from initial inertial forces.
2L p (t)
L
L
{
load length
Z (t) Rigid uniform bar Total mass = m
Pulley: Total mass = m (uniform over area) Inextensible massless cable
c
k FIGURE P8-2
p(t)
Rigid massless bar Rigid uniform bar (Total mass = m)
k
k Z(t) L
c L 2
L
L 2
FIGURE P8-3
x
Z (t)
v(x,t) =
(x) Z(t)
p=
L
load length
Uniform column m=
mass length
EI = flexural rigidity
FIGURE P8-4
x 2
8 ft
(x) = 1
200 ft
1
cos
x
2L
Concrete stack: density = 150 lb/ft3 E = 3 10 6 lb/in2 wall thickness = 8 in
0 18 ft
FIGURE P8-5
{
m1
m = mass length EI
x
Assumed shape L 2
v(x)
L 2
FIGURE P8-6
v1(0) = 1
m1 k1
v2(0) = 2 3
v3(0) = 1 3
m2 k2 m3 k3
kips sec2/in kips/in kips sec2/in kips/in kips sec2/in kips/in FIGURE P8-8
CHAPTER
FORMULATION OF THE MDOF EQUATIONS OF MOTION
p(x,t)
{
1 m(x) EI(x) v1 (t)
i
2
v 2 (t)
N
vi (t)
vN (t)
FIGURE 9-1 Discretization of a general beam-type structure.
CHAPTER
EVALUATION OF STRUCTURALPROPERTY MATRICES
~ v 1 = f 11
~ f 21 p1 = 1
~ f 12
~ fN 1
~ f i1
~ vi = fi 2 ~ f 22 p2 = 1
~ fN 2 FIGURE 10-1 Definition of flexibility influence coefficients.
p 1 = k 11 p 2 = k 21
pi 1 = k i 1
pN 1 = k N 1
pi = k i 2
pN = k N 2
v1 =1
p 2 = k 22 p 1 = k 12 v2 =1
FIGURE 10-2 Definition of stiffness influence coefficients.
Load system a: p1 a
Load system b: p2 a
p1 b
p3 a
p2 b
p3 b
Deflections b:
Deflections a:
v3 b v1 a
v2 a
v3 a
FIGURE 10-3 Two independent load systems and resulting deflections.
v1 b
v2 b
v(x)
EI(x)
a
b L x
va
v1 = 1
a
1 (x)
v3 = 1 3 (x)
FIGURE 10-4 Beam deflections due to unit nodal displacements at left end.
v(x) = va v3
a
=1
v1
1 (x)
v1
3 (x)
pa = k 13
FIGURE 10-5 Beam subjected to real rotation and virtual translation of node.
v2
v3 4EI v1
L
EI
EI 2L (a) k 21 = 2EI (3L) L3
k 31 = 2EI (3L) L3 2EI
k 11 = v1 = 1
L3
(6)(2)
(b) k 22 = =
2EI L3 2EI L3
(2L 2 ) +
2(4EI ) (2L) 3
(2)(2L) 2
k 32 =
(6L 2 )
2(4EI) (2L) 3
k 12 =
(2L) 2 =
2EI L3
2EI L3
(2L 2 )
(3L)
v2 = 1
(c)
FIGURE E10-1 Analysis of frame stiffness coefficients: (a) frame properties and degrees of freedom; (b) forces due to displacement v1 = 1; (c) forces due to rotation v2 = 1.
0
1
2
i
m0 a
m1 a
m2 c
mi k
m1 b
m2 b
mi l
m1
m2
mi
N
mN
FIGURE 10-6 Lumping of mass at beam nodes.
v(x)
m(x) b
a L v(x) = v1 v¨3
1 (x)
v1
va
¨ =1 a
Inertial force fI (x) pa = m 13
FIGURE 10-7 Node subjected to real angular acceleration and virtual translation.
v2
v3 1.5 m
L
v1
m
1.5 mL
m11 = 4 mL
1.5 mL
0.5 mL
0.5 mL
m 2L
0.5 mL
0.5 mL
(a) m 21 =
(b)
mL mL (22L) = (11L) 420 210
m 31 =
mL (11L) 210
(Axial motion of girder)
mL (156)(2) + (1.5 m)(2L) 420 mL = (786) 210
m 11 = v¨ 1 = 1
(c) m 22 = =
(1.5 m)(2L) mL (4L2) + (4)(2L)2 420 420 mL (26L2) 210
m 32 =
(1.5 m)(2L) mL ( 3)(2L2)2 = ( 18L2) 420 210
m 12 =
mL mL (22L) = (11L) 210 420
v¨ 2 = 1 (d) FIGURE E10-2 Analysis of lumped- and consistent-mass matrices: (a) uniform mass in members; (b) lumping of mass at member ends; (c) forces due to acceleration v¨1 = 1 (consistent); (d ) forces due to acceleration v¨2 = 1 (consistent).
( , )
=
1
1
( )=
1(
)
1
A. Der Kiureghian, “Structural Response to Stationary Excitation,” loc. cit.
A. Der Kiureghian, “Structural Response to Stationary Excitation,” loc. cit.
v1 m
v3 L x
L
JG = ¾2 EI each member 3
L m= v2 x = 0.05
z Fixed
=
1 0 0
0 1 0 4.59 4.83 14.56
y FIGURE E26-7 3-DOF system subjected to rigid-base translation.
0 0 m 1
f=
=
13 3
3 25
12
-3
12 L 3 - 3 ¾¾ 6EI 19
0.731 0.271 1.000 - 0.232 1.000 - 0.242 1.000 0.036
- 0.787
1.3
z (t) z x (t)
1.2 max max
Eq. (26-129)
1.1 Eq. (26-127)
1.0
0
0.2
0.4 B=
0.6 z y (t)
max
z x (t)
max
0.8
1.0
FIGURE 26-13 Statistical approach versus 30% rule in combining two components of horizontal response.
Story mass: mi = 24 kips sec 2 ft Total column stiffness: EI = 4 10 6 kips ft2 L = 480 ft
Story height: h = 12 ft
FIGURE P26-1 Uniform shear building.
z L m
EI
m va vb
2L
EI w = 0.377 1.25
EI mL3
EI = 100 1/sec 2 mL3 z
F = 1.00 1.00 0.85 - 2.35
FIGURE P26-2 2-DOF plane frame.
CHAPTER
DETERMINISTIC EARTHQUAKE RESPONSE: INCLUDING SOIL-STRUCTURE INTERACTION
y
D
x L
FIGURE 27-1 Rigid rectangular basemat of a large structure.
(1985) and (1988), both published by Prentice-Hall, Inc., Englewood Cliffs, N. J.
1.0 0.8 0.6 0.4 0.2 0
0
4
2
3 4
5 4
3 2
7 4
D 2 D = Va FIGURE 27-2 factor as a function of frequency and apparent wave velocity.
2
9 4
5 2
Structure Structure and soil response degrees of freedom v(t)
Interface
Free soil boundary
Foundation medium
Rigid base, input degrees of freedom v¨g (t) FIGURE 27-3 Finite-element model of combined structure and supporting soil.
, McGraw-Hill Book Company, New York (1975), pp. 584–588.
v(t)
h
c
k 2
k 2
m 0 , J0
y
Fixed reference
m, J h
c
k 2
vg (t)
Elastic half-space
v0 (t) M0 (t)
v(t)
M0 (t)
Rigid massless plate Elastic half-space
(a) (b) FIGURE 27-4 Lumped SDOF elastic system on rigid mat foundation.
Substructure No. 1
k 2
t
vz
m, J
I
(t) I
(t)
Substructure No. 2
A. S. Veletsos and Y. T. Wei, “Lateral and Rocking Vibrations of Footings,” , ASCE, Vol. 97, 1971. J. E. Luco and R. A. Westman, “Dynamic Response of Circular Footings,” , ASCE, Vol. 97, (EM5), 1971.
p(t) = exp (i t) T(t) = exp (i t) M(t) = exp (i t) v(t) = exp (i t) 2R Half-space p, G, v
FIGURE 27-5 Rigid massless circular plate on half-space.
1.0
v=0
= 3 G R(a 0 ) 16GR 3
v) G R(a 0 ) 4GR
1.0
1 3
0.5
= (1
v =1 2
1 3 1 2
(a) 2
4 a0 = R
6
1 3
0
1 2 1 3
0
1 2
0.5
(c) 0
2
0.7 0.6
Torsion
0.5 0.3 0.2 0.1 0
(b) 0
Torsion
2
4 a0 = R
Vs
v=0
1.0
8
0.8
= 3 (1 v) G I (a 0 ) 8GR 3a 0
0
= 3 (1 v) G R(a 0 ) 8GR 3
= (2 v) G (a 0 ) 8 GR a 0
0.5
Vertical translation
= 3 G I (a 0 ) 16GR 3a 0
v=0 1.0
I
= (2 v) G R(a 0 ) 8GR
= (1 v) G I (a 0 ) 4GR a 0
Vertical translation
0.9
Lateral translation
4 a0 = R
6
0
1.0
6
Rocking
v=0 0.5
(d)
8
1 3
1 3
0
0
2
Vs
1 2
1 2
4 a0 = R
= (2 v) G I (a 0 ) 8 GR 2a 0
= (2 v) G R(a 0 ) 8 GR 2
0.3 v=0
0.2
1 3
0.1
1 2
v=0
0 0.1 0.2
(e)
1 2
1 3
Coupled lateral translation and rocking
0
FIGURE 27-6 Rigid massless circular plate impedances.
2
4 a0 = R
8
Vs
6 Vs
8
6 Vs
8
FIGURE 27-7 Example structures for soil-structure interaction analysis.
(a)
(b)
(d)
(c)
FIGURE 27-8 Substructures Nos. 1 and 2 for the systems shown in Fig. 26-7.
nA
nb
nc
Soil
Near field
nd
nd Far field Half-space with surface cavity
FIGURE 27-9 Modeling of foundation full half-space.
H. B. Seed, R. T. Wong, I. M. Idriss, and K. Tokimatsu, “Moduli and Damping Factors for Dynamic Analysis for Cohesionless Soils,” University of California, Berkeley, Earthquake Engineering Research Center, Report No. EERC 84-14, 1984. B. O. Hardin and V. P. Drnevich, “Shear Modulus and Damping in Soils: Design Equations and Curves,” , Vol. 98, No. SM7, July, 1972. J. Lysmer, T. Udaka, C. F. Tsai, and H. B. Seed, “Flush — A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problem,” University of California, Berkeley, Earthquake Engineering Research Center, Report No. EERC 75–30, 1975. I. Katayama, C. H. Chen, and J. Penzien, “Near-Field Soil-Structure Interaction Analysis Using Nonlinear Hybrid Modeling,” Proc. SMIRT Conference, Anaheim, Ca., 1989.
(0, t) (0, t)
(0, t) (0, t)
y, v
v(0, t)
w (0, t)
z, w
cp =
Column Foundation half-space
cs =
G
D
(b)
(a) FIGURE 27-10 Substructure No. 2 as a uniform half-space having viscous boundary elements as its equivalent.
(0, t) Layer H1
vs 1 ,
(0, t)
(0, t) 1,
vs 2 ,
y1, v1 z 1, w 1
1 2,
v1 (0, t)
(0, t)
y2, v2 z 2, w2
2
cs
w1 (0, t) Half-space Shear-beam column
kp
ks
cp
(b)
(a) FIGURE 27-11 Substructure No. 2 as a uniform layer on a uniform half-space having viscous and spring boundary elements as its equivalent.
mg Rigid hammer
vh m
m
Cushion spring: k
wm (t)
wm (t)
w(0, t)
w(0, t)
m w¨m (t) k cz =
z, w(z,t) Vp =
E
AE Vp
Pile: A, E, N (0, t) (a)
FIGURE E27-1 Hammer-cushion-pile system.
(b)
Axial force, kips 0
0
200
400
600
N (z, 0.005067)
Distance from top of pile, ft
10 Vp = (1.15)(10 5) 20 in sec
30
40 48.6
50 z
FIGURE E27-2 Axial-force distribution in concrete pile 0.00507 sec after initial hammer impact with cushion.
T. J. Tzong, S. Gupta, and J. Penzien, “Two-Dimensional Hybrid Modeling of Soil-Structure Interaction,” Report No. UC-EERC 81/11, Earthquake Engineering Research Center, University of California, Berkeley, August, 1981. T. J. Tzong and J. Penzien, “Hybrid-Modeling of a Single Layer Half-Space System in Soil-Structure Interaction,” , Vol. 14, 1986.
u
1.00
iIt
Cn n = R n iIn
1 exp (i t)
0.75 Ct t = Rt
1 exp (i t)
1.00
0.50
a a t
= (Ct t G)exp(i t) It
0.25
0.75 0.50
a a un = (Cn n G)exp(i t)
0.25
0
0
1.00
2.00 a0
0
3.00
0
iIr
a a ull = (Cll G )exp(i t)
0.40
Crr = Rr
iIl Cll = Rl
1.5
0.5
0.30
Il 1.0 a0
a
3.00
a
Vs
u rr = (Cr r a 2G)exp(i t) Rr a a
0.20
Ir
0.10
Rl 0
2.00 a0
Vs
0.50
0
1.00
a
2.0
1.0
In
Rn
Rt
2.0
0
3.0
0
1.00 a0
Vs
a
2.00
3.00
Vs
FIGURE 27-12 Compliances of infinite rigid massless strip of width 2a; G = shear modulus, Vs = shear-wave velocity.
Cavity R
S
Surface layer of depth H
H SR
vs 1 = G 1 1 , G1
1
CL Sym.
Half-space vs 2 = G 2 2, G2
2
FIGURE 27-13 Continuous far-field impedance functions Sp and SR along halfcylindrical cavity surface.
30
Real part constant form
(
14 0
(
R0
3
6
25
9
44
12
15
9
6
9
Imaginary part freq. form
30
(Vs 2 Vs 1) 2 = 3.0 10.0
45 0
6
0
(Vs 2 Vs 1) 2 = 1.0 3.0 10.0
28
3
0
15
Imaginary part constant form
R G1)
R G1)
60
4
3
R1
20
8
(
(
10
R G1)
0
R1
10
Real part freq. form
19
R0
R G1)
20
3
6
60
9
3
0
R
b0
b0
Vs1
FIGURE 27-14 Parameters defining impedance SR along half-cylindrical cavity surface.
0
0
0
0
1
0
0
1
0
1 1
1
1
0
1
1
R
Vs1
10
R G1)
0 5
(
(Vs 2 Vs 1) = 1.0 3.0 10.0
0
3
6
10
9
R G1) (
(
(Vs 2 Vs 1) 2 = 3.0 10.0
0
20
6 1 8
4 3
Imaginary part constant form
13
11
3
6
9
Imaginary part freq. form
13 6
1
R G1)
20
0
1
2
10 15
Real part freq. form
18
(
R G1)
5
0
25
Real part constant form
1 8
0
3
6 b0
9
15
0
R
Vs1
3
6 b0
9
R
Vs1
FIGURE 27-15 Parameters defining impedance S along half-cylindrical cavity surface.
A. S. Veletsos and Y. T. Wei, “Lateral and Rocking Vibrations of Footings,” loc. cit. J. E. Luco and R. A. Westman, “Dynamic Response of Circular Footings,” loc. cit. A. S. Veletsos and V. V. D. Nair, “Torsional Vibration of Foundations,” Structural Research at Rice, Report No. 19, Department of Civil Engineering, Rice University, June, 1973.
1
1
2
S
2
Mode i
S SR
Tributary area i
FIGURE 27-16 Far-field impedances over the hemispherical cavity surface in spherical co-ordinates.
E. Kausel, “Forced Vibrations of Circular Footings on Layered Media,” MIT Research Report R74-11, Mass. Inst. of Tech., Cambridge, Mass., 1974. J. E. Luco, “Impedance Functions for a Rigid Foundation on a Layered Medium,” , Vol. 31, No. 2, 1979. S. Gupta, T. W. Lin, J. Penzien, and C. S. Yeh, “Three-Dimensional Hybrid Modeling of Soil-Structure Interaction,” ., Vol. 10, No. 1, Jan. – Feb., 1982.
Real part
Imaginary part
6
30
4
R G)
20
2
R
0
(
(
R
R G)
8
0
3
6
10
0 9 0 Normal component
3
6
9
3
6
9
6
9
15
2
10
R G)
4
(
R G)
6
2
(
0 0
3
6
5
0 9 0 Tangential component
8 15 R G)
4
10
(
R G)
6
0
(
2 0
3
6 p0 =
R
Vs
5
0 9 0 Circumferential component
FIGURE 27-17 Far-field impedance functions over the hemispherical cavity surface.
3 p0 =
R
Vs
F
4 1
2
F
3
(a) Free-field deformations FIGURE 27-18 Modeling for cross-section racking analyses.
(b) SSI deformations
J. Penzien, C. H. Chen, W. Y. Jean, and Y. J. Lee, “Seismic Analysis of Rectangular Tunnels in Soft Ground,” Proceedings of the Tenth World Conference on Earthquake Engineering, Madrid, Spain, July, 1992.
X, U
Y, V y, v Vf f
V( X,
t)
Direction of wave propagation
X
x Axis of tunnel
V(X, t)
u(x, t) = V(X, t) sin
x, u
v(x, t) = V(X, t) cos
FIGURE 27-19 Shear wave moving in the X direction at velocity Vf f .
S. Gupta, T. W. Lin, J. Penzien, and C. S. Yeh, “Three-Dimensional Hybrid Modeling of Soil-Structure Interaction,” loc. cit.
Tunnel
no joints
a a max x
Tunnel
with joints
j
j
a
mp
L
x FIGURE 27-20 Tunnel axial strains with and without joints.
CHAPTER
STOCHASTIC STRUCTURAL RESPONSE
G. W. Housner, “Behavior of Structures During Earthquakes,” 1959.
, Vol. 85, No. EM-4, October,
G. N. Bycroft, “White Noise Representation of Earthquakes,” Proc. Paper 2434, , Vol. 86, EM2, April, 1960.
Sv , ft /sec
1.5
Housner’s design velocity spectra Bycrott’s velocity spectra for S 0 = 0.0063 ft 2 sec3 ( and freq.) =0
1.0
= 0.02 = 0.05
0.5 = 0.10
0
0.5
1.0
Multiplication factors given by Housner
1.5 Period
2.0
El Centro El Centro Olympia Taft
1940 1934 1949 1952
2.5
2.7 1.9 1.9 1.6
3.0
T, sec
FIGURE 28-1 Mean extreme values of pseudo-relative velocity for linear SDOF systems (stationary white-noise excitation).
J. Penzien and S. C. Liu, “Nondeterministic Analysis of Nonlinear Structures Subjected to Earthquake Excitations,” Proc. 4th World Conf. Earthquake Eng., Santiago, Chile, Vol. I, Sec. A-1, January, 1969.
Average of 50 artificial earthquakes
=0
1
= .02 = .05 = .10
0
1
2
3
Sv , ft /sec
Sv , ft /sec
G. Housner’s design spectra
1
=0 = .02 = .05 = .10
0
Period T , sec
1 2 Period T , sec
(a)
(b)
FIGURE 28-2 Mean extreme values of pseudo-relative velocity for linear SDOF systems (filtered stationary white-noise excitation).
3
2
B = Vy W = 4 vt
m= W g
V
v
d
vmax
v
V Vy
k
c
= v max vy V ve max Vy
vy
Vy
C
E (a)
(b)
vy
D (c)
FIGURE 28-3 Nonlinear SDOF models.
TABLE 28-1 Case No.
Structural type *
Period T sec
Damping ratio,
1 2 3 4 5 6 7 8 9 10 11 12
E EP SD E EP SD E EP SD E EP SD
0.3 0.3 0.3 0.3 0.3 0.3 2.7 2.7 2.7 2.7 2.7 2.7
0.02 0.02 0.02 0.10 0.10 0.10 0.02 0.02 0.02 0.10 0.10 0.10
Elastic-plastic
SD
*E
Elastic
EP
Strength ratio, B
Yield displ. vy in
0.10 0.10
0.088 0.088
0.10 0.10
0.088 0.088
0.048 0.048
3.42 3.42
0.048 0.048
3.42
Stiffness degrading
B k
Vy
vg
A
F
v
k
vy gT 2
v
Displacement v max,
in
FIGURE 28-4 Probability distributions for extreme values of relative displacement. 0
2.5
0.001
2.0 3.0 4.0 5.0
0.300 0.500 0.700 0.800
0.001
1.001 1.40 2.0
3.0 4.0 5.0
10
0.300 0.500 0.700 0.800 0.900
0.0
1.0
2.0
Probability distribution P( v)
0.100
P( v) = exp [ exp ( v)] where v = ( v max u)
1.10
(a)
3.0
0
0.5
10
15
20
25
30
35
40
45
50
Return period, no. of earthquakes
(b)
2.0
0.950
d
Reduced variate, v
1.0
0.900
1
2
3
4
5
6
7
20
in
Reduced variate, v
0.0
Probability distribution P( v)
0.100
10
Ductility demand
5.0
1.40
P( v) = exp [ exp ( v)] where v = ( v max u)
1.10
max,
7.5
1.001
Displacement v
10.0
12.5
15.0
17.5
20.0
22.5
25.0
Return period, no. of earthquakes
3.0
0.950
10
20
30
40
50
20
Ductility demand d
E. J. Gumbel and P. G. Carlson,
, op. cit.; E. J. Gumbel, , op. cit.
(1)
(4) (3)
0.5
0
0
0.5
E [vmax ] , 30
(6)
1.0
(5)
T = 0.3 sec
(1) (2) (3) (4) (5) (6)
E, E, EP, EP, SD, SD,
= 0.02 = 0.10 = 0.02 = 0.10 = 0.02 = 0.10
E [vmax ] , T0
(2)
E [vmax ] , T0
E [vmax ] , 30
1.0
1.0
(2) (6) (1)
(5)
0.5
0
(4)
0
(3)
T = 2.7 sec
(1) (2) (3) (4) (5) (6)
E, E, EP, EP, SD, SD,
= 0.02 = 0.10 = 0.02 = 0.10 = 0.02 = 0.10
0.5
T0 30
T0 30
(a)
(b)
FIGURE 28-5 Duration effect of stationary process on mean peak response of linear and nonlinear structures.
1.0
M. Murakami and J. Penzien, “Nonlinear Response Spectra for Probabilistic Seismic Design and Damage Assessment of Reinforced Concrete Structures,” Univ. of Calif., Berkeley, Earthquake Engineering Research Center, Report No. 75–38, 1975. H. Umemura et. al., , Giko-do, Tokyo, Japan, 1973 (in Japanese).
C
B
pB y
Original skeleton curve
k1 ky 1 1
pB y pB c = 3
pB c O
vBy = vmin
vBc
P
(a)
A
1
vBc
k2
1
ky 1
O
1 D
k2
vBy
vm ax 2 vB y = vmax vmin
pB c
B
pB y pB y
Q
R
1
O
vBy S
(b)
k1
pB c
A 1
O vB c
vB c
k1 k2
C
ky
1 D
vB y
P
Skeleton curve after first yielding R
Q
FIGURE 28-6 Trilinear stiffness-degrading hysteretic model.
P. C. Jennings, G. W. Housner, and N. C. Tsai, “Simulated Earthquake Motions,” loc. cit.
0
EI(x) m(x) L
x,
FIGURE P23-2 Cantilever member of Prob. 23-3.
CHAPTER
SEISMOLOGICAL BACKGROUND
N. M. Newmark and E. Rosenblueth, Englewood Cliffs, N. J., 1971.
, Prentice-Hall Inc.,
N Strike
Strike-slip fault (left-lateral)
Dip
Reverse fault Normal fault
FIGURE 24-2 Definition of fault orientation, and of the basic types of fault displacement. [Adapted from Earthquake by Bruce A. Bolt, W. H. Freeman and Company 1988.]
P-wave
(a)
Compressions
Dilatations
S-wave
Double amplitude
(b)
Wavelength
Love wave
(c) Rayleigh wave
(d) FIGURE 24-3 Diagram illustrating the forms of ground motion near the ground surface in four types of earthquake waves. [From Bruce A. Bolt, Nuclear Explosions and Earthquakes: The Parted Veil (San Francisco: W. H. Freeman and Company. Copyright © 1976).]
Earthquake focus Reflection at the surface Mantle Core
Seismograph station
Refraction at the core FIGURE 24-4 Paths of some P-type earthquake waves from the focus.
Crust Mantle Outer core (liquid)
Inner core (solid)
FIGURE 24-5 Zonation of the earth’s interior. The crust, which includes continents at the surface of the earth, rests on the mantle. The mantle, in turn, rests on the core. The outer core is liquid, but the inner core is solid. [After W. J. Kauffman, Planets and Moons, W. H. Freeman and Company, New York, 1979.]
Asiatic Plate
Juan De Fuca Plate San Andreas Fault Philippine Sea Plate Cocos Pacific Plate Plate Australian Plate
European Plate
North American Plate
Caribbean Plate
Nazca Plate
Antarctic Plate
FIGURE 24-6 Simplified map of the Earth’s crustal plates.
South American Plate
African Plate
Greenland Iceland British Isles M
i d-
A
Kilometers
tla
T IC
ntic R idge
AT L A N
OCEA
N
Axis of ridge
100 150
0
50
0
30 60 Miles
90
FIGURE 24-7 Magnetic-anomaly pattern of the North Atlantic sea floor. Symmetrical striping is revealed by measurement of the strength of the magnetic field at many locations from a ship. The position of the area represented in the lower diagram is shown in the map above. [From A. Cox et al., "Reversals of the Earth’s Magnetic Field." Copyright © 1967 by Scientific American, Inc. All rights reserved.]
0.3
Acceleration Acceleration of gravity
0.2 0.1 0 0.1 0.2 0.3 0
5
10
15
20
Time, sec FIGURE 24-15 Accelerogram from El Centro earthquake, May 18, 1940 (NS component).
25
9˚40’ W
9˚30’ W
VI Tarhazout VII VIII
Tamarhout
30˚30’ N
Ait Lamine
Kasbah
Anza
Agadir Atlantic Ocean
Scale of Miles 3
4
5
r
2
Sous
VI
Ben Sergao Inezgane ve
1
Epicenter VIII Yachech Talbordit New City VII Industrial Zone (South)
Ri
0
IX
Ait Melloul
30˚20’ N
FIGURE 24-16 Isoseismal map of Agadir earthquake, 1960 (Modified Mercalli intensity scale).
CHAPTER
FREE-FIELD SURFACE GROUND MOTIONS
v t (t) v(t)
Reference axis
m k 2
vg (t)
c
k 2
FIGURE 25-1 Basic SDOF dynamic system.
D. E. Hudson, “Response Spectrum Techniques in Engineering Seismology,” , Earthquake Engineering Research Institute, Berkeley, CA, 1956. D. E. Hudson, “Some Problems in the Application of Spectrum Techniques to Strong Motion Earthquake Analysis,” , Vol. 52, No. 2, April, 1962.
10
S p v ( , T ), ft /sec
8 =0
6 4 2 0
= 0.2
= 0.02 = 0.4
0
0.5
1.0
1.5
2.0
2.5
3.0
Undamped natural period, T FIGURE 25-2 Pseudo-velocity response spectra, El Centro, California earthquake, May 18, 1940 (NS component).
250.0
Sp v ( , T ), cm sec
100.0 50.0 25.0
10.0 5.0 2.5 0.05
0.1
0.5
1
5
10
Undamped natural period, T FIGURE 25-3 Pseudo-velocity response spectra for El Centro, California earthquake, May 18, 1940 (NS component).
1
1
G. W. Housner, “Spectrum Intensities of Strong Motion Earthquakes,” , Earthquake Engineering Research Institute, Los Angeles, 1952.
H. B. Seed and I. M. Idriss, “Ground Motions and Soil Liquefaction During Earthquakes,” Monograph published by the Earthquake Engineering Research Institute, 1982. N. M. Newmark and W. J. Hall, “Earthquake Spectra and Design,” Monograph published by the Earthquake Engineering Research Institute 1982. G. W. Housner, “Design Spectrum,” Englewood Cliffs, N. J., 1970.
, Chapter 5, Ed. R. L. Wiegel, Prentice-Hall,
G. W. Housner, “Properties of Strong Ground Motion Earthquakes,” , Vol. 45, No. 3, July, 1955.
ad (t)
Ground surface
d
h
Shear beam model
Soil
Bedrock b
c
a b (t)
a c (t)
FIGURE 25-4 The shear beam model used for soil response analyses.
H. B. Seed, C. Ugas, and J. Lysmer, “Site-Dependent Spectra for Earthquake Resistant Design,” , Vol. 66, No. 1, February, 1976. J. Penzien, “Statistical Nature of Earthquake Ground Motions and Structural Response,” Proc. U.S.Southeast Asia Symposium on Engineering for Natural Hazards Protection, Manila, Philippines, September, 1977.
4
Spectral acceleration Maximum ground acceleration
Total number of records analysed: 104
Spectra for 5% damping
3 Soft to medium clay and sand
15 records
Deep cohesionless soils (> 250 ft)
2
Stiff soil conditions (< 150 ft) Rock
30 records 31 records
28 records
1
0
0
0.5
1.0
1.5 Period T
2.0
2.5
3.0
seconds
FIGURE 25-5 Average pseudo-acceleration spectra for different site conditions (by Seed et al.).
4 Total number of records analysed: 104
Spectral acceleration Maximum ground acceleration
Spectra for 5% damping
3
Soft to medium clay and sand
15 records
Deep cohesionless soils (> 250 ft)
30 records
Stiff soil conditions (< 150 ft)
31 records
2
Rock
28 records
1
0
0
0.5
1.0
1.5 Period T
2.0
2.5
seconds
FIGURE 25-6 84 percentile pseudo-acceleration spectra for different site conditions (by Seed et al.).
3.0
1.0 R = 1.60 T = 1.0 sec = 0.05 R = 0.73 Deep cohesionless soils
P (R)
0.8 0.6 0.4 0.2 0
0
1.0
2.0
3.0
4.0
5.0
R 1.0 R = 0.61 T = 1.0 sec = 0.05 R = 0.40 Rock
P (R)
0.8 0.6 0.4 0.2 0
0
0.5
1.0
1.5 R
G. W. Housner, “Design Spectrum,” Englewood Cliffs, N. J., 1970.
2.0
2.5
FIGURE 25-7 Gumbel Type I probability distribution functions for response ratio R for two soil types.
, Chapter 5, Ed. R. L. Weigel, Prentice-Hall,
C. H. Loh, J. Penzien, and Y. B. Tsai, “Engineering Analysis of SMART-1 Array Accelerograms,” , Vol. 10, 1982.
3
K. W. Campbell, “Near-Source Attenuation of Peak Horizontal Acceleration,” , Vol. 76, No. 6, December, 1981. C. A. Cornell, “Engineering Seismic Risk Analysis,” Vol. 58, No. 5, October, 1968. A. Der Kiureghian and A. H-S. Ang, “A Fault-Rupture Model for Seismic Risk Analysis,” , Vol. 67, No. 4, August, 1977.
,
M. G. Bonilla, “A Review of Recently Active Faults in Taiwan,” U.S. Department of the Interior, Geological Survey, Open-File Report 75-41, 1975.
N. M. Newmark and W. J. Hall, “Earthquake Spectra and Design,” loc. cit.
d
al
e
S
S
Lo
g
sc
a
Lo a
a
d
Sv Log scale
vv
a
g
sc
al
e
d
v a
f2 = 4 f1 f3 = 10 f1 0.1
f1 f=
2
Log scale
f2 f3
60
FIGURE 25-8 Design response spectrum.
W. J. Hall, “Observations on Some Current Issues Pertaining to Nuclear Power Plant Seismic Design,” , North-Holland Publishing Co., Vol. 69, 1982. B. Mohraz, “A Study of Earthquake Response Spectra for Different Geological Conditions,” , Vol. 66, No. 3, June, 1976.
Spectral acceleration Maximum ground acceleration
Soil profile type S 3 (soft)
3
Soil profile type S 2 (medium) Soil profile type S 1 (hard)
2.35
2
Spectra for 5% damping
1.62
0.92 0.77
1
0
0.5
1.0
2.056
1.370
1.147
0.64
0
0.43
0.47
1.5 Period T
2.0
2.5
3.0
seconds
FIGURE 25-9 ATC-3 normalized response spectra recommended for use in building code.
Applied Technology Council (ATC), “Tentative Provisions for the Development of Seismic Regulations for Buildings,” ATC Publication ATC3-06, NBS Special Publication 510, and NSF Publication 78-8, 1978.
t( el
isp
la
cc at
D
er )
10
1
(g
10
n
3
io
= 0.05 2
10 0
1
10
10
10
1
1
0
10
10
10
10
en m ce
A
Velocity (m/sec)
10 1
4
cm
)
10 2
1
10 0
2
10
Period (sec)
10 1
10 2
FIGURE 25-10 Normalized response spectrum curves for = 0.05 using eight accelerograms recorded in Taipei, Taiwan during the earthquake of November 14, 1986.
t( en
10
m ce la
le
Di
ce
sp
Ac
10
n(
1
10
tio
3
ra g)
= 0.05 2
10 0
10
1
10
1
10
10
10
1
0
10
Velocity (m/sec)
10 1
4
cm
)
10 2
1
10 0
2
10
Period (sec)
10 1
10 2
FIGURE 25-11 Normalized design response spectrum curve for = 0.05 representing Taipei, Taiwan, soft soil conditions.
N. M. Newmark, J. A. Blume, and K. K. Kapur, “Seismic Design Spectra for Nuclear Power Plants,” , Vol. 99, No. PO2, November, 1973. U. S. Atomic Energy Commission, Regulatory Guide 1.60, “Design Response Spectra for Seismic Design of Nuclear Power Plants,” Revision 1, December, 1973.
HORIZONTAL
100
1
2
5
7
0
10
1.
0
10
)
0.
01
(in
0
t 10 n 0 . e me lac
Ac
sp
ce
10
Di
le 0. rati 10 on
0
(g
)
1.
Velocity (in/sec)
0
0.5
10
10
Percent critical damping
10
1000
0. 01 0
0.1
1.0
0.1
10
100
Frequency (cps) VERTICAL
1
2
5
7
0
10
(g
10 n 0 . e me lac
Ac
sp
ce
Di
le 0. rati 10 on
0
10
1.
)
1.
0
10
in
01
0
t( )
0.
Velocity (in/sec)
0
0.5
100
10
10
Percent critical damping 10
1000
0. 01 0
0.1
0.1
1.0
10
Frequency (cps)
100
FIGURE 25-12 NRC smooth design response spectrum curves (mean 1 levels) normalized to 1g peak ground acceleration.
H. B. Seed, C. Ugas, and J. Lysmer, “Site-Dependent Spectra for Earthquake Resistant Design,” loc. cit. R. K. McGuire, “A Simple Model for Estimating Fourier Amplitude Spectra of Horizontal Ground Acceleration,” , Vol. 68, No. 3, June, 1978. M. D. Trifunac, “Preliminary Empirical Model for Scaling Fourier Amplitude Spectra of Strong Ground Acceleration in Terms of Earthquake Magnitude, Source-to-Site Distance, and Recording Site Conditions,” , Vol. 66, No. 4, August, 1976. D. M. Boore, “Stochastic Simulation of High-Frequency Ground Motions Based on Seismological Models of the Radiated Spectra,” , Vol. 73, No. 6, December, 1983.
f (t) 1 t t1 t1
2
e
c (t
t 2)
t2
t
FIGURE 25-13 Intensity function f (t) for nonstationary process a(t).
P. Ruiz and J. Penzien, “Probabilistic Study of Behavior of Structures during Earthquakes,” University of California, Berkeley, Earthquake Engineering Research Center, Rept. 69-3, 1969. P. C. Jennings, G. W. Housner, and N. C. Tsai, “Simulated Earthquake Motions,” Rept., Earthquake Engineering Research Laboratory, California Institute of Technology, April, 1968. G. W. Housner, “Design Spectrum,” loc. cit.
1 2 2 1
1
2 2 2 2
2
J. L. Bogdanoff, J. E. Goldberg, and M. C. Bernard, “Response of a Simple Structure to a Random Earthquake-Type Disturbance,” , Vol. 51, No. 2, April, 1961. T. Kubo and J. Penzien, “Time and Frequency Domain Analyses of Three-Dimensional Ground Motions, San Fernando Earthquake,” University of California, Berkeley, Earthquake Engineering Research Center, Report No. 76-6, March, 1976. K. Kanai, “Semi-empirical Formula for the Seismic Characteristics of the Ground,” University of Tokyo, Bull., Earthquake Research Institute, Vol. 35, pp. 309-325, 1957; H. Tajimi, “A Statistical Method of Determining the Maximum Response of a Building Structure during an Earthquake,” Proc. 2nd World Conference on Earthquake Engineering, Tokyo and Kyoto, Vol. II, pp. 781-798, July, 1960.
H(i )
2
H1 (i ) H2 (i )
1
FIGURE 25-14 Absolute value of combined filter function.
1.5
Acceleration, g
1.0 0.5 0 0.5 1.0 1.5
0
6
12
18
24
30
Time, sec FIGURE 25-15 Synthetic accelerogram adjusted to be compatible with smooth design spectrum.
K. Lilhanand and W. S. Tseng, “Development and Application of Realistic Earthquake Time Histories Compatible with Multiple-Damping Design Spectra,” Proceedings of the Ninth World Conference on Earthquake Engineering, Tokyo/Kyoto Japan (Vol. II), August 29, 1988. M. Watabe, “Characteristics and Synthetic Generation of Earthquake Ground Motions,” Proc., Canadian Earthquake Engineering Conference, July, 1987.
200 100 80 60
Velocity, in sec
40 20 10 8 6 4
.2 .1
.06 .08 .1
.2
.4
.6 .8 1
2
4
6 8 10
20
Period, sec FIGURE 25-16 Smooth design response spectrum and response spectrum for adjusted synthetic accelerogram; = 0.05.
1.5 1g peak acceleration
Acceleration, g
1.0 0.5 0 0.5 1.0 1.5
0
6
12
18
24
30
18
24
30
Time, sec FIGURE 25-17 Normalized accelerogram
Taft California N21˚E, 1952.
1.5
Acceleration, g
1.0 0.5 0 0.5 1.0 1.5
0
6
12 Time, sec
FIGURE 25-18 Taft accelerogram adjusted to be compatible with smooth design spectrum.
200
Spectrum compatible Taft accelerogram
100 80 60
Velocity, in sec
40 20 Normalized Taft accelerogram
10 8 6 4 2 1 .04 .06 .08 .1
.2
.4
.6 .8 1
2
4
6 8 10
20
Period, sec FIGURE 25-19 Smooth design response spectrum and response spectra for normalized Taft accelerogram and adjusted Taft accelerogram; = 0.05.
0
0
0
J. Penzien and M. Watabe, “Simulation of 3-Dimensional Earthquake Ground Motions,” , Vol. (1974), and , Vol. 3, No. 4, April-June, 1975.
North
Acceleration cm sec2 100 80 60 40
20 Interval t1-sec t2-sec 1 2 3 4 5 6
2 6 10 14 18 22
6 10 14 18 22 26
1
2
4
3
E East
6
Tokachi-Oki, Japan (Hachinoe Station) May 16, 1968.
5
FIGURE 25-20 Directions of major principal axis of ground motion Tokachi-Oki, Japan, earthquake (Hachinoe Station) May 16, 1968.
0
0
0 0 0
0
0
0
0 0
C. H. Loh and J. Penzien, “Identification of Wave-Types, Directions, and Velocities Using SMART-1 Strong Motion Array Data,” Proc., 8th World Conference on Earthquake Engineering, San Francisco, Ca., July 21-28, 1984. T. Harada, “Probabilistic Modeling of Spatial Variation of Strong Earthquake Ground Displacements,” Proc., 8th World Conference on Earthquake Engineering, San Francisco, Ca., July 21-28, 1984. N. A. Abrahamson and B. A. Bolt, “The Spatial Variation of the Phasing of Seismic Strong Ground Motion,” , Vol. 75, No. 5, October, 1985. R. S. Harichandran and E. H. Vanmarke, “Stochastic Variation of Earthquake Ground Motion in Space and Time,” , February, 1986.
H. Hao, C. S. Oliveira, and J. Penzien, “Multiple-Station Ground Motion Processing and Simulation Based on SMART-1 Array Data,” , North-Holland, Amsterdam, 1989. E. Samaras, M. Shinozuka, and A. Tsurui, “Time Series Generation Using the Auto-Regressive MovingAverage Model,” Technical Report, Department of Civil Engineering, Columbia University, New York, May, 1983.
CHAPTER
DETERMINISTIC EARTHQUAKE RESPONSE: SYSTEMS ON RIGID FOUNDATIONS
v t (t) v(t)
Reference axis
m
c k 2
fS 2
vg (t)
k 2
fS 2
FIGURE 26-1 Lumped SDOF system subjected to rigid-base translation.
x
Z(t)
L
Reference axis
v t (x, t)
v(x, t) =
(x) Z (t)
EI(x) m(x)
vg (t)
FIGURE 26-2 Generalized SDOF system with rigid-base translation.
x
x
f S (x) L
f S (x) = h
2
m(x) v(x) h
Vh
Mh (b)
V0
M0 (a)
FIGURE 26-3 Elastic-force response of generalized SDOF system: (a) base forces; (b) section forces.
Z (t) Assumed shape: v(x, t) = (1
cos
x
2L
) Z (t)
m = 0.02 kips sec 2 / ft 2 L = 100 ft
EI = 14
10 5 kips ft 2
= 5%
vg (t)
FIGURE E26-1 SDOF idealization of uniform cantilever column.
v1t x
Reference axis
m1 m2 m3
vit
mi
vi
mN
vg ( t )
FIGURE 26-4 Discretized MDOF system with rigid-base translation.
fS 1
m1 m2
fS 2
fS 3
m3 mi
mN v0
fS i
fS N M0
FIGURE 26-5 Elastic forces in lumped MDOF system.
kip sec2/in
=
kips /in kips sec2/in
kips /in kips sec2/in
n
=
n
=
kips /in
1.000 0.644 0.300 4.58 9.83 14.57
1.000 0.601 0.676
1.00 2.57 ; 2.47
sec 1; Tn =
2.566 1.254 2.08
FIGURE E26-2 Building frame and its vibration properties.
kips sec2/in
1.37 0.639 0.431
=
21.0 0 0
0 96.6 0
sec ; M n =
n
0 0 212.4
1.801 2.455 23.10
= 0.05
sec
2
kips sec2/in ;
max
max
max
max
Reference axis
m2
m1
vg (t)
v1
m3
m4
v3
v4
v2
FIGURE 26-6 General lumped MDOF system with rigid-base translation.
m
2m
EI
v1
L EI L
v2 m = 0.01 kips sec /ft 2
EI = 1 kip/ft L3
= 0.05 vg (t)
m=
=
3 0
0 2
0.431 1.000
FIGURE E26-3 Two-DOF frame and its vibration properties.
10
2
1.000 0.646
kips sec2/ft
=
k=
6 7
8 3
0.302 0 0 2.84
3 2
kips/ft
10 2 sec
2
y v2 v v3 1
x
FIGURE 26-7 Rigid slab subjected to base translation.
Direction of ground motion
Rigid slab Total mass m = 0.5 kips sec2/ft
v3
v2 L
k= v1
EI = 5 kips/ft L3
(each column)
L e ak qu n rth ta tio a E xci e
L
A
C
B
FIGURE E26-4 Slab supported by three columns.
k
k 31 = 2 k 1
k 21 =
2k
k k
k
k 1
1
k
k
k 11 = 4 k
1
k
(a)
(b)
FIGURE E26-5 Evaluation of stiffness coefficients for v1 = 1: (a) displacement v1 = 1 and resisting column forces; (b) column forces and equilibrating stiffness coefficients.
v¨ 2 = 1 mL ¾ 6 1¾ L
m 32 = - m2¾ m 22 = 2¾ m 3
mL ¾ 6
m¾ 2
m¾ 2
m¾ 2
m¾ 2
m 12 = - 1¾ m 6
1 (a)
(b)
FIGURE E26-6 Evaluation of mass coefficients for v¨ 2 = 1. (a) Acceleration v¨ 2 = 1 and resisting M inertial forces; (b) slab inertial forces and equilibrating mass coefficients.
Applied Technology Council (ATC), “Tentative Provisions for the Development of Seismic Regulations for Buildings,” loc. cit.
x3 m2
v2
x4 m3
m4 v3
v1 h2
v4
m1 h1 =1 Reference axis
FIGURE 26-8 Lumped MDOF system with rigid-base rotation.
v1s
v3s
Reference axis
m1, J1 v2s
v4s
v3
v1
m2, J2 v2 v4
h1 h2
g
FIGURE 26-9 Tower with lumped masses having rotational inertias subjected to rigid-base rotation.
Nodal response degrees of freedom vt
Support input degrees of freedom vg
FIGURE 26-10 General finite element earthquake response model.
fs (v) fs y
1 k fs y
vy
v FIGURE 26-11 Elastic-plastic force-displacement relation.
N. M. Newmark and W. J. Hall, “Earthquake Spectra and Design,” loc. cit.
fs fs, el
fs fs, el
0 < f < 0.3 Hz k
fs y = fs, elpl
1
fs y = fs, elpl v vel = velpl = vg
vy
(a)
max
0.3 < f < 2 Hz 1 k v vel = velpl
vy
(b)
FIGURE 26-12 Elastic and elastic-plastic force-displacement relations.
fs fs, el
fs ,y = fs, elpl vg
max
2 < f < 8 Hz 1 k vy
vel velpl
(c)
v
2 2
A. Der Kiureghian, “Structural Response to Stationary Excitation,” loc. cit.
A. Der Kiureghian, “Structural Response to Stationary Excitation,” loc. cit.
v1 m
v3 L x
L
JG = ¾2 EI each member 3
L m= v2 x = 0.05
z Fixed
=
1 0 0
0 1 0 4.59 4.83 14.56
y FIGURE E26-7 3-DOF system subjected to rigid-base translation.
0 0 m 1
f=
=
13 3
3 25
12
-3
12 L 3 - 3 ¾¾ 6EI 19
0.731 0.271 1.000 - 0.232 1.000 - 0.242 1.000 0.036
- 0.787
1.3
z (t) z x (t)
1.2 max max
Eq. (26-129)
1.1 Eq. (26-127)
1.0
0
0.2
0.4 B=
0.6 z y (t)
max
z x (t)
max
0.8
1.0
FIGURE 26-13 Statistical approach versus 30% rule in combining two components of horizontal response.
Story mass: mi = 24 kips sec 2 ft Total column stiffness: EI = 4 10 6 kips ft2 L = 480 ft
Story height: h = 12 ft
FIGURE P26-1 Uniform shear building.
z L m
EI
m va vb
2L
EI w = 0.377 1.25
EI mL3
EI = 100 1/sec 2 mL3 z
F = 1.00 1.00 0.85 - 2.35
FIGURE P26-2 2-DOF plane frame.
CHAPTER
DETERMINISTIC EARTHQUAKE RESPONSE: INCLUDING SOIL-STRUCTURE INTERACTION
y
D
x L
FIGURE 27-1 Rigid rectangular basemat of a large structure.
(1985) and (1988), both published by Prentice-Hall, Inc., Englewood Cliffs, N. J.
1.0 0.8 0.6 0.4 0.2 0
0
4
2
3 4
5 4
3 2
7 4
D 2 D = Va FIGURE 27-2 factor as a function of frequency and apparent wave velocity.
2
9 4
5 2
Structure Structure and soil response degrees of freedom v(t)
Interface
Free soil boundary
Foundation medium
Rigid base, input degrees of freedom v¨g (t) FIGURE 27-3 Finite-element model of combined structure and supporting soil.
, McGraw-Hill Book Company, New York (1975), pp. 584–588.
v(t)
h
c
k 2
k 2
m 0 , J0
y
Fixed reference
m, J h
c
k 2
vg (t)
Elastic half-space
v0 (t) M0 (t)
v(t)
M0 (t)
Rigid massless plate Elastic half-space
(a) (b) FIGURE 27-4 Lumped SDOF elastic system on rigid mat foundation.
Substructure No. 1
k 2
t
vz
m, J
I
(t) I
(t)
Substructure No. 2
A. S. Veletsos and Y. T. Wei, “Lateral and Rocking Vibrations of Footings,” , ASCE, Vol. 97, 1971. J. E. Luco and R. A. Westman, “Dynamic Response of Circular Footings,” , ASCE, Vol. 97, (EM5), 1971.
p(t) = exp (i t) T(t) = exp (i t) M(t) = exp (i t) v(t) = exp (i t) 2R Half-space p, G, v
FIGURE 27-5 Rigid massless circular plate on half-space.
1.0
v=0
= 3 G R(a 0 ) 16GR 3
v) G R(a 0 ) 4GR
1.0
1 3
0.5
= (1
v =1 2
1 3 1 2
(a) 2
4 a0 = R
6
1 3
0
1 2 1 3
0
1 2
0.5
(c) 0
2
0.7 0.6
Torsion
0.5 0.3 0.2 0.1 0
(b) 0
Torsion
2
4 a0 = R
Vs
v=0
1.0
8
0.8
= 3 (1 v) G I (a 0 ) 8GR 3a 0
0
= 3 (1 v) G R(a 0 ) 8GR 3
= (2 v) G (a 0 ) 8 GR a 0
0.5
Vertical translation
= 3 G I (a 0 ) 16GR 3a 0
v=0 1.0
I
= (2 v) G R(a 0 ) 8GR
= (1 v) G I (a 0 ) 4GR a 0
Vertical translation
0.9
Lateral translation
4 a0 = R
6
0
1.0
6
Rocking
v=0 0.5
(d)
8
1 3
1 3
0
0
2
Vs
1 2
1 2
4 a0 = R
= (2 v) G I (a 0 ) 8 GR 2a 0
= (2 v) G R(a 0 ) 8 GR 2
0.3 v=0
0.2
1 3
0.1
1 2
v=0
0 0.1 0.2
(e)
1 2
1 3
Coupled lateral translation and rocking
0
FIGURE 27-6 Rigid massless circular plate impedances.
2
4 a0 = R
8
Vs
6 Vs
8
6 Vs
8
FIGURE 27-7 Example structures for soil-structure interaction analysis.
(a)
(b)
(d)
(c)
FIGURE 27-8 Substructures Nos. 1 and 2 for the systems shown in Fig. 26-7.
nA
nb
nc
Soil
Near field
nd
nd Far field Half-space with surface cavity
FIGURE 27-9 Modeling of foundation full half-space.
H. B. Seed, R. T. Wong, I. M. Idriss, and K. Tokimatsu, “Moduli and Damping Factors for Dynamic Analysis for Cohesionless Soils,” University of California, Berkeley, Earthquake Engineering Research Center, Report No. EERC 84-14, 1984. B. O. Hardin and V. P. Drnevich, “Shear Modulus and Damping in Soils: Design Equations and Curves,” , Vol. 98, No. SM7, July, 1972. J. Lysmer, T. Udaka, C. F. Tsai, and H. B. Seed, “Flush — A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problem,” University of California, Berkeley, Earthquake Engineering Research Center, Report No. EERC 75–30, 1975. I. Katayama, C. H. Chen, and J. Penzien, “Near-Field Soil-Structure Interaction Analysis Using Nonlinear Hybrid Modeling,” Proc. SMIRT Conference, Anaheim, Ca., 1989.
(0, t) (0, t)
(0, t) (0, t)
y, v
v(0, t)
w (0, t)
z, w
cp =
Column Foundation half-space
cs =
G
D
(b)
(a) FIGURE 27-10 Substructure No. 2 as a uniform half-space having viscous boundary elements as its equivalent.
(0, t) Layer H1
vs 1 ,
(0, t)
(0, t) 1,
vs 2 ,
y1, v1 z 1, w 1
1 2,
v1 (0, t)
(0, t)
y2, v2 z 2, w2
2
cs
w1 (0, t) Half-space Shear-beam column
kp
ks
cp
(b)
(a) FIGURE 27-11 Substructure No. 2 as a uniform layer on a uniform half-space having viscous and spring boundary elements as its equivalent.
mg Rigid hammer
vh m
m
Cushion spring: k
wm (t)
wm (t)
w(0, t)
w(0, t)
m w¨m (t) k cz =
z, w(z,t) Vp =
E
AE Vp
Pile: A, E, N (0, t) (a)
FIGURE E27-1 Hammer-cushion-pile system.
(b)
Axial force, kips 0
0
200
400
600
N (z, 0.005067)
Distance from top of pile, ft
10 Vp = (1.15)(10 5) 20 in sec
30
40 48.6
50 z
FIGURE E27-2 Axial-force distribution in concrete pile 0.00507 sec after initial hammer impact with cushion.
T. J. Tzong, S. Gupta, and J. Penzien, “Two-Dimensional Hybrid Modeling of Soil-Structure Interaction,” Report No. UC-EERC 81/11, Earthquake Engineering Research Center, University of California, Berkeley, August, 1981. T. J. Tzong and J. Penzien, “Hybrid-Modeling of a Single Layer Half-Space System in Soil-Structure Interaction,” , Vol. 14, 1986.
u
1.00
iIt
Cn n = R n iIn
1 exp (i t)
0.75 Ct t = Rt
1 exp (i t)
1.00
0.50
a a t
= (Ct t G)exp(i t) It
0.25
0.75 0.50
a a un = (Cn n G)exp(i t)
0.25
0
0
1.00
2.00 a0
0
3.00
0
iIr
a a ull = (Cll G )exp(i t)
0.40
Crr = Rr
iIl Cll = Rl
1.5
0.5
0.30
Il 1.0 a0
a
3.00
a
Vs
u rr = (Cr r a 2G)exp(i t) Rr a a
0.20
Ir
0.10
Rl 0
2.00 a0
Vs
0.50
0
1.00
a
2.0
1.0
In
Rn
Rt
2.0
0
3.0
0
1.00 a0
Vs
a
2.00
3.00
Vs
FIGURE 27-12 Compliances of infinite rigid massless strip of width 2a; G = shear modulus, Vs = shear-wave velocity.
Cavity R
S
Surface layer of depth H
H SR
vs 1 = G 1 1 , G1
1
CL Sym.
Half-space vs 2 = G 2 2, G2
2
FIGURE 27-13 Continuous far-field impedance functions Sp and SR along halfcylindrical cavity surface.
30
Real part constant form
(
14 0
(
R0
3
6
25
9
44
12
15
9
6
9
Imaginary part freq. form
30
(Vs 2 Vs 1) 2 = 3.0 10.0
45 0
6
0
(Vs 2 Vs 1) 2 = 1.0 3.0 10.0
28
3
0
15
Imaginary part constant form
R G1)
R G1)
60
4
3
R1
20
8
(
(
10
R G1)
0
R1
10
Real part freq. form
19
R0
R G1)
20
3
6
60
9
3
0
R
b0
b0
Vs1
FIGURE 27-14 Parameters defining impedance SR along half-cylindrical cavity surface.
0
0
0
0
1
0
0
1
0
1 1
1
1
0
1
1
R
Vs1
10
R G1)
0 5
(
(Vs 2 Vs 1) = 1.0 3.0 10.0
0
3
6
10
9
R G1) (
(
(Vs 2 Vs 1) 2 = 3.0 10.0
0
20
6 1 8
4 3
Imaginary part constant form
13
11
3
6
9
Imaginary part freq. form
13 6
1
R G1)
20
0
1
2
10 15
Real part freq. form
18
(
R G1)
5
0
25
Real part constant form
1 8
0
3
6 b0
9
15
0
R
Vs1
3
6 b0
9
R
Vs1
FIGURE 27-15 Parameters defining impedance S along half-cylindrical cavity surface.
A. S. Veletsos and Y. T. Wei, “Lateral and Rocking Vibrations of Footings,” loc. cit. J. E. Luco and R. A. Westman, “Dynamic Response of Circular Footings,” loc. cit. A. S. Veletsos and V. V. D. Nair, “Torsional Vibration of Foundations,” Structural Research at Rice, Report No. 19, Department of Civil Engineering, Rice University, June, 1973.
1
1
2
S
2
Mode i
S SR
Tributary area i
FIGURE 27-16 Far-field impedances over the hemispherical cavity surface in spherical co-ordinates.
E. Kausel, “Forced Vibrations of Circular Footings on Layered Media,” MIT Research Report R74-11, Mass. Inst. of Tech., Cambridge, Mass., 1974. J. E. Luco, “Impedance Functions for a Rigid Foundation on a Layered Medium,” , Vol. 31, No. 2, 1979. S. Gupta, T. W. Lin, J. Penzien, and C. S. Yeh, “Three-Dimensional Hybrid Modeling of Soil-Structure Interaction,” ., Vol. 10, No. 1, Jan. – Feb., 1982.
Real part
Imaginary part
6
30
4
R G)
20
2
R
0
(
(
R
R G)
8
0
3
6
10
0 9 0 Normal component
3
6
9
3
6
9
6
9
15
2
10
R G)
4
(
R G)
6
2
(
0 0
3
6
5
0 9 0 Tangential component
8 15 R G)
4
10
(
R G)
6
0
(
2 0
3
6 p0 =
R
Vs
5
0 9 0 Circumferential component
FIGURE 27-17 Far-field impedance functions over the hemispherical cavity surface.
3 p0 =
R
Vs
F
4 1
2
F
3
(a) Free-field deformations FIGURE 27-18 Modeling for cross-section racking analyses.
(b) SSI deformations
J. Penzien, C. H. Chen, W. Y. Jean, and Y. J. Lee, “Seismic Analysis of Rectangular Tunnels in Soft Ground,” Proceedings of the Tenth World Conference on Earthquake Engineering, Madrid, Spain, July, 1992.
X, U
Y, V y, v Vf f
V( X,
t)
Direction of wave propagation
X
x Axis of tunnel
V(X, t)
u(x, t) = V(X, t) sin
x, u
v(x, t) = V(X, t) cos
FIGURE 27-19 Shear wave moving in the X direction at velocity Vf f .
S. Gupta, T. W. Lin, J. Penzien, and C. S. Yeh, “Three-Dimensional Hybrid Modeling of Soil-Structure Interaction,” loc. cit.
Tunnel
no joints
a a max x
Tunnel
with joints
j
j
a
mp
L
x FIGURE 27-20 Tunnel axial strains with and without joints.
CHAPTER
STOCHASTIC STRUCTURAL RESPONSE
G. W. Housner, “Behavior of Structures During Earthquakes,” 1959.
, Vol. 85, No. EM-4, October,
G. N. Bycroft, “White Noise Representation of Earthquakes,” Proc. Paper 2434, , Vol. 86, EM2, April, 1960.
Sv , ft /sec
1.5
Housner’s design velocity spectra Bycrott’s velocity spectra for S 0 = 0.0063 ft 2 sec3 ( and freq.) =0
1.0
= 0.02 = 0.05
0.5 = 0.10
0
0.5
1.0
Multiplication factors given by Housner
1.5 Period
2.0
El Centro El Centro Olympia Taft
1940 1934 1949 1952
2.5
2.7 1.9 1.9 1.6
3.0
T, sec
FIGURE 28-1 Mean extreme values of pseudo-relative velocity for linear SDOF systems (stationary white-noise excitation).
J. Penzien and S. C. Liu, “Nondeterministic Analysis of Nonlinear Structures Subjected to Earthquake Excitations,” Proc. 4th World Conf. Earthquake Eng., Santiago, Chile, Vol. I, Sec. A-1, January, 1969.
Average of 50 artificial earthquakes
=0
1
= .02 = .05 = .10
0
1
2
3
Sv , ft /sec
Sv , ft /sec
G. Housner’s design spectra
1
=0 = .02 = .05 = .10
0
Period T , sec
1 2 Period T , sec
(a)
(b)
FIGURE 28-2 Mean extreme values of pseudo-relative velocity for linear SDOF systems (filtered stationary white-noise excitation).
3
2
B = Vy W = 4 vt
m= W g
V
v
d
vmax
v
V Vy
k
c
= v max vy V ve max Vy
vy
Vy
C
E (a)
(b)
vy
D (c)
FIGURE 28-3 Nonlinear SDOF models.
TABLE 28-1 Case No.
Structural type *
Period T sec
Damping ratio,
1 2 3 4 5 6 7 8 9 10 11 12
E EP SD E EP SD E EP SD E EP SD
0.3 0.3 0.3 0.3 0.3 0.3 2.7 2.7 2.7 2.7 2.7 2.7
0.02 0.02 0.02 0.10 0.10 0.10 0.02 0.02 0.02 0.10 0.10 0.10
Elastic-plastic
SD
*E
Elastic
EP
Strength ratio, B
Yield displ. vy in
0.10 0.10
0.088 0.088
0.10 0.10
0.088 0.088
0.048 0.048
3.42 3.42
0.048 0.048
3.42
Stiffness degrading
B k
Vy
vg
A
F
v
k
vy gT 2
v
Displacement v max,
in
FIGURE 28-4 Probability distributions for extreme values of relative displacement. 0
2.5
0.001
2.0 3.0 4.0 5.0
0.300 0.500 0.700 0.800
0.001
1.001 1.40 2.0
3.0 4.0 5.0
10
0.300 0.500 0.700 0.800 0.900
0.0
1.0
2.0
Probability distribution P( v)
0.100
P( v) = exp [ exp ( v)] where v = ( v max u)
1.10
(a)
3.0
0
0.5
10
15
20
25
30
35
40
45
50
Return period, no. of earthquakes
(b)
2.0
0.950
d
Reduced variate, v
1.0
0.900
1
2
3
4
5
6
7
20
in
Reduced variate, v
0.0
Probability distribution P( v)
0.100
10
Ductility demand
5.0
1.40
P( v) = exp [ exp ( v)] where v = ( v max u)
1.10
max,
7.5
1.001
Displacement v
10.0
12.5
15.0
17.5
20.0
22.5
25.0
Return period, no. of earthquakes
3.0
0.950
10
20
30
40
50
20
Ductility demand d
E. J. Gumbel and P. G. Carlson,
, op. cit.; E. J. Gumbel, , op. cit.
(1)
(4) (3)
0.5
0
0
0.5
E [vmax ] , 30
(6)
1.0
(5)
T = 0.3 sec
(1) (2) (3) (4) (5) (6)
E, E, EP, EP, SD, SD,
= 0.02 = 0.10 = 0.02 = 0.10 = 0.02 = 0.10
E [vmax ] , T0
(2)
E [vmax ] , T0
E [vmax ] , 30
1.0
1.0
(2) (6) (1)
(5)
0.5
0
(4)
0
(3)
T = 2.7 sec
(1) (2) (3) (4) (5) (6)
E, E, EP, EP, SD, SD,
= 0.02 = 0.10 = 0.02 = 0.10 = 0.02 = 0.10
0.5
T0 30
T0 30
(a)
(b)
FIGURE 28-5 Duration effect of stationary process on mean peak response of linear and nonlinear structures.
1.0
M. Murakami and J. Penzien, “Nonlinear Response Spectra for Probabilistic Seismic Design and Damage Assessment of Reinforced Concrete Structures,” Univ. of Calif., Berkeley, Earthquake Engineering Research Center, Report No. 75–38, 1975. H. Umemura et. al., , Giko-do, Tokyo, Japan, 1973 (in Japanese).
C
B
pB y
Original skeleton curve
k1 ky 1 1
pB y pB c = 3
pB c O
vBy = vmin
vBc
P
(a)
A
1
vBc
k2
1
ky 1
O
1 D
k2
vBy
vm ax 2 vB y = vmax vmin
pB c
B
pB y pB y
Q
R
1
O
vBy S
(b)
k1
pB c
A 1
O vB c
vB c
k1 k2
C
ky
1 D
vB y
P
Skeleton curve after first yielding R
Q
FIGURE 28-6 Trilinear stiffness-degrading hysteretic model.
P. C. Jennings, G. W. Housner, and N. C. Tsai, “Simulated Earthquake Motions,” loc. cit.
0
