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458
APPENDIX A Similitude for Model Tests in a 1-g Gravitational Field (Iai, 1989)
Geometric Scaling Factor 8
Soil Density Scaling Factor 1
Model Soil Shear Wave Velocity (Vs)m 100
Prototype Soil Shear Wave Velocity (Vs) 282.8
Soil Strain Scaling Factor = / [(Vs)p/(Vs)m]2
1
prototype:model
x length 8
density of saturated soil p 1
strain of soil 1
t time ()0.5
2.83
0 strain of soil due to creep, temperature, etc. 1
total stress of soil 8
' effective stress of soil 8
D tangent modulus of soil p/ 8
Ks bulk modulus of the solid grains of soil p/ 8
p pressure of pore water and/or external water 8
k permeability of soil ()0.5
/p 2.83
u displacement of soil and/or structure 8
u velocity of soil and/or structure ()0.5
2.83
u acceleration of soil and/or structure 1 1
w average displacement of pore water relative to the soil skeleton 8
w rate of pore water flow ()0.5
2.83
n porosity of soil 1 1
Kf bulk modulus of pore water and/or external water p/ 8
EI flexural rigidity (per unit breadth of the beam) 4p/ 4096
EA longitudinal rigidity (per unit breadth of the beam) 2
p/ 64 inclination of the beam 1
M bending moment of the beam (per unit breadth of the beam) 3 512
S shear force of the beam (per unit breadth of the beam) 2 64
F axial force of the beam (per unit breadth of the beam) 2p 64
f density of pore water and/or external water p 1
b density of the beam ( mass per unit length and breadth of the beam) 8
T traction acting on the soil specified on the boundary 8
u displacement of the soil and/or the beam specified on the boundary 8
p pressure of pore water and/or external water specified on the boundary 8
w average displacement of pore water, on the boundary, relative to the soil skeleton 8
inclination of the beam specified at the boundary 1
M bending moment of the beam specified at the boundary (per unit breadth) 3 512
S shear force of the beam specified at the boundary (per unit breadth) 2 64
F axial force of the beam specified at the boundary (per unit breadth) 2p 64
i hydraulic gradient of external water specified at the boundary p 1
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459
APPENDIX B Model Pile Design Spreadsheet
Geometric Scaling Factor: 8
Prototype Input Parameters
Prototype Pile OD: 16 inchesPrototype Pile Wall: 0.50 inches
Prototype Pile Length: 44 feet
Prototype Soil Shear Strength: 1000 psf
Prototype Soil Vs: 1000 fps
E Steel: 29000 ksi
E Concrete: 4000 ksi
% Concrete EI Contribution: 50%
Prototype Pile Computed Properties
Prototype Pile L/D Ratio: 33
Prototype Pile d/t Ratio: 32Prototype Epile/Gsoil: 1392
Prototype EIpile/Esoil*D^4: 96
Area Steel: 0.1691 ft^2
Steel Moment of Inetria: 0.0353 ft^4
Steel Flexural Rigidity EI: 147405 k-ft^2
Area Concrete: 1.2272 ft^2
Concrete Moment of Inertia: 0.1198 ft^4
Concrete EI: 34515 k-ft^2
Composite Concrete/Steel EI: 181920 k-ft^2
Total Mass/ft length: 266.93 lbs/ft
Prototype First Mode Period: 0.7386 seconds
Model Input Parameters
Model Pile OD: 2 inches
Model Pile Wall: 0.028 inches
Model Pile E: 10000 ksi
Model Pile Density: 40.00 pcf
Model Soil Vs: 100 fps
Model Pile Computed Properties
Model Pile Cross Sectional Area: 0.0012 ft^2
Model Pile Mass/ft length 0.0482 lbs/ft
Model Pile Moment of Inertia: 4.0673E-06 ft^4 Target % difference
Model Pile EI: 5.8568 k-ft^2 5.5517 5%
Model Pile L/D Ratio: 36.0 33.0 9%
Model Pile d/t Ratio: 71.4 32.0 123%
Model Pile First Mode Period: 0.0273 seconds 0.2611 90%
Model Epile/Gsoil: 3840 1392 176%
Model EIpile/Esoil*D^4: 101 96 5%
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APPENDIX C Analysis of Seismic Response of Cylindrical Tank
Container Properties:radius R, height H, wall thickness h, soil unit weight, soil mass m, depth z:
R .3.5 ft H .7 ft h .0.064 in .94lb
ft3
m ... R2
H =m 12.661 ton z ..,.0 ft .1 ft H
Container dynamic respose modeled as flexible wall tank with three response components:hydrodynamic convective, hydrodynamic impulsive, and hydrostatic.
Convective Response: modal frequency fc, modal mass mc, modal height hc
from the zeroes of the Bessel function derivative: 1.841
fc .1
.2 ..
g
Rtanh .
H
R=fc 0.654 Hz
mc ...
2 m
2
1
R. H
tanh . HR
=mcm
0.227
hc H .R
tanh .
H
.2 R=
hc
H0.742
Response Maxima: Peak ground acceleration xmax, amplification factorc, pseudoacceleration Ac:
xmax .0.8 g c 1 Ac .c xmax =Ac 0.8 g kip .1000 lbf
Wall pressure pc, base shear Qc, base moment Mc, normal stress c, shear stress c, hoop stress c:
Qc .mc Ac =Qc 4.6 kip
Mc ..mc hc Ac =Mc 23.883 .kip ftCc( )z .
2
2
1
cosh . zR
cosh .H
Rc
Mc
.. R2
h
=c 808.047 psi
pc( )z ...Cc( )z R Ac cQc
.. R h=c 544.692 psi
c( )z .pc( )zR
h c( )z
psi
50.517
57.668
81.146
127.597
210.173
352.252
594.058
.1.004 103
pc( )z
psi
0.0770.088
0.124
0.194
0.32
0.537
0.905
1.53
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Impulsive Response: modal frequency fi, effective modal mass mi, effective modal height hi,peak ground acceleration xmax, amplification factori, pseudoacceleration Ai :
fi .1 Hz mi .0.763 m hi .0.422 H i 2.5 Ai .i xmax =Ai 2 g
Wall pressure pi, base shear Qi, base moment Mi, normal stress i, shear stress i, hoop stress i:
Ci( )z ..( )H z0.133
fte
.0.186z
ftQi .mi Ai =Qi 38.643 kip
Mi ..mi hi Ai =Mi 114.15 .kip ft
pi( )z ...Ci( )z R Ai
iMi
.. R2
h
=i 3.862 103
psi
pi( )z
psi
4.254
4.3924.408
4.247
3.837
3.081
1.855
0
iQi
.. R h=i 4.576 10
3psi
i( )z.
pi( )z
R
h
i( )z
psi
.2.792 103
.2.882 103
.2.893 103
.2.787 103
.2.518 103
.2.022 103
.1.217 103
0
Hydrostatic Response: wall pressure ph, hoop stress h:
ph( )z ..( )H z g
ph( )z
psi
4.569
3.917
3.264
2.611
1.958
1.3060.653
0
h( )z .ph( )zR
h
h( )z
psi
.2.999 103
.2.57 103
.2.142 103
.1.714 103
.1.285 103
856.771
428.385
0
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Total Dynamic Response: Conservatively sum components (ABSSUM):Wall pressure p, base shear Q, base moment M, normal stress , shear stress , hoop stress :
p( )z pc( )z pi( )z ph( )zQ Qc Qi =Q 43.242 kipp( )z
psi8.901
8.396
7.796
7.053
6.115
4.923
3.413
1.53
( )z c( )z i( )z h( )z M Mc Mi =M 138.033 .kip ft
c i = 4.67 103
psi
c i = 5.121 103
psi
0 2 4 6 8 100
2
4
6
8
0
z
ft
zft
0
,p( )z
psi
ph( )z
psi
0 1000 2000 3000 4000 5000 60000
2
4
6
8
z
ft
z
ft
,( )z
psi
h( )z
psi
( )z
psi
.5.841 103
.5.51 103
.5.116 103
.4.628 103
.4.013 103
.3.231 103
.2.24 103
.1.004 103
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