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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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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