Transcript Fundamentals-Modeling-Properties-Performance

Vibrations
of
Machine Foundations
Richard P. Ray, Ph.D., P.E.
Civil and Environmental Engineering
University of South Carolina
USC
Thanks To:
Prof. Richard D. Woods, Notre Dame Univ.
Prof. F.E. Richart, Jr.
USC
Topics for Today
Fundamentals
 Modeling
 Properties
 Performance

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Foundation Movement
Z
Y
θ
φ
X
ψ
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Design Questions (1/4)
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How Does It Fail?
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Static Settlement
Dynamic Motion Too Large (0.02 mm is large)
Settlements Caused By Dynamic Motion
Liquefaction
What Are Maximum Values of Failure?
(Acceleration, Velocity, Displacement)
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Velocity Requirements
Massarch (2004) "Mitigation of Traffic-Induced Ground Vibrations"
Fundamentals-Modeling-Properties-Performance
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Design Questions (2/4)

What Are Relations Between Loads And
Failure Quantities

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Loading -Machine (Periodic), Impluse, Natural
Relations Between Load, Structure, Foundation,
Soil, Neighboring Structures
Generate Model: Deterministic or Probabilistic
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Design Questions (3/4)
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How Do We Measure What Is Necessary?
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Full Scale Tests
Prototype Tests
Small Scale Tests (Centrifuge)
Laboratory Tests (Specific Parameters)
Numerical Simulation
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Design Questions (4/4)
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What Factor of Safety Do We Use?
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Does FOS Have Meaning
What Happens After There Is Failure
Loss of Life
 Loss of Property
 Loss of Production
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Purpose of Project, Design Life, Value
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r -2
r -2 r -0.5
+
-
+
Rayleigh wave
Vertical
Horizontal
component component
+
Shear
wave
+
Relative
amplitude
-
r -1
+
+
Shear
window
r
r -1
Waves
Fundamentals-Modeling-Properties-Performance
Wave Type
Percentage of
Total Energy
Rayleigh
67
Shear
26
Compression
7
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Modeling Foundations

Lumped Parameter (m,c,k) Block System


Impedance Functions


Function of Frequency (ω), Layers
Boundary Elements (BEM)

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Parameters Constant, Layer, Special
Infinite Boundary, Interactions, Layers
Finite Element/Hybrid (FEM, FEM-BEM)

Complex Geometry, Non-linear Soil
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Lumped Parameter
P  Po sin( t )
r
m
Gνρ
m
c
k
mz  cz  kz  P0 sin(t )
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SDOF
Mag 

Adynamic
Astatic
1
 
1   
  n 
Fundamentals-Modeling-Properties-Performance
2 2
2
 

  2 D 
  n 
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Lumped Parameter System
Z
mz z  cz z  kz z  P0 sin( t )
Cz
Kz
Iψ
m
ψ
k
n 
m
Kx
X
Cx
D  c ccr ccr  2 k m
Kψ
Cψ/2
Cψ/2
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Lumped Parameter Values
Mode
Stiffness k
Mass Ratio
mˆ
m
Damping
Ratio, D
Fictitious
Mass
Vertical
Horizontal
4Gr
1 
8Gr
2 
m(2   )
8r 3
0.288
mˆ 1/ 2
0.095 m
mˆ
m(1   )
4 r 3
0.425
mˆ 1 / 2
0.27 m
mˆ
Rocking
8Gr 3
3(1   )
3I (1   )
8r 5
0.15
(1  mˆ )mˆ 1/ 2
0.24 I x
mˆ
Torsion
16Gr 3
3
I
r 5
0.50
1  2 mˆ
0.24 I z
mˆ
D=c/ccr G=Shear Modulus ν=Poisson's Ratio r=Radius
ρ=Mass Density Iψ,Iθ=Mass Moment of Inertia
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Mass Ratio
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Design Example 1
VERTICAL COMPRESSOR
Unbalanced Forces
•Vertical Primanry
= 7720 lb
•Vertical Secondary
= 1886 lb
•Horzontal Primary
=
104 lb
•Horizontal Secondary =
0 lb
Operating Speed
=
450 rpm
Wt Machine + Motor = 10 900 lb
Soil Properties
DESIGN CRITERION:
Smooth Operation At Speed
Shear Wave Velocity Vs = 680 ft/sec
Shear Modulus,
G = 11 000 psi
Velocity <0.10 in/sec
Density,
γ = 110 lb/ft3
Displacement < 0.002 in
Poisson's Ratio,
ν = 0.33
Jump to Chart
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Q0
(1  )Q0 0.667(7720 1885)
Azs 
 0.002" 

kz
4Gr
4 11 000 r
r  72.8"  6.07 '
Try a 15 x 8 x 3 foundation block, Area = 120 ft2 and r = 6.18 ft
Weight = 54,000 lb Total Weight = 54 000 + 10 900 = 64 900
mˆ 
(1  ) W g 0.67  64 900

 0.42
3
3
4r g  4 1106.18
0.425
 1 
 0.66 M z  1.0 

mˆ
 2D 
Az dynamic  Az static  0.002"
D
Jump to Figure
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Design Example - Table Top
18'
Q0=400 lb
34'
W=550 000 lb
Iψ=2.88 x 106 ft-lb-sec2
DESIGN CRITERION
ψ
18'
11'
0.20 in/sec Horizontal Motion at
Machine Centerline
Soil Properties
Ax = 0.0015 in. from combined
rocking and sliding
Shear Wave Velocity Vs = 770 ft/sec
Speed = 160 rpm
Shear Modulus,
G = 14 000 psi
Slower speeds, Ax can be larger
Density,
γ = 110 lb/ft3
Poisson's Ratio,
ν = 0.33
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Horizontal Translation Only
2  m
 0.38
3


8 r
0.288
Q Q 2 
D  1/ 2  0.465  Magx  1.0 Ax static  0  0
 3.0 105 in
mˆ
kx
8 Gr
Equivanlent r 
4cd

4 18 34
 13.96 ft mˆ 
Rocking About Point "O"
3
3
16
cd
16

17

9
Equivalent r  4
4
 12.0 ft
3
3
  120 rpm  12.5 rad / sec
k
8Gr 8  (14 000144) 12.04
2.90108
8
k 

 2.9010 lb / ft n 

 10 rad / sec
6
2 
2  0.33
I
2.8810
3(1  ) I
3(0.67) 2.88106
mˆ  

 0.83
110
8
r 5
8
5
(12.04)
32.2
0.15
D 
 0.09  Mag  5.6 Static Mom ent M o  40018  7200ft  lbs.
(1  mˆ  ) mˆ 
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Static AngularDeflection  s 
M o 7200 3(0.67) 0.50

 6 rad
8
k
2.9 10
10
0.50
4
(
18

12
)

1
.
0

10
in
6
10
At Resonance 5.6(1.0 104 )  5.6 104 in.
HorizontalMotion  Axs   s  h 
6
5
Damping = 9%
Mag
4
3
2
1
0
0
0.5
1
1.5
2
2.5
OmegaRatio
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Impedance Methods

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Based on Elasto-Dynamic Solutions
Compute Frequency-Dependent Impedance
Values (Complex-Valued)
Solved By Boundary Integral Methods
Require Uniform, Single Layer or Special Soil
Property Distribution
Solved For Many Foundation Types
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Impedance Functions
P  Poeit  Po cos( t )  i sin(t ) 
Sz


Rz
2K
Sz 
 K  iC  K STATIC  k ( )  i C 
DSOIL 
Az



Soil Damping
Radiation Damping
Jump Wave
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Impedance Functions
a0   r

G

r
Vs
ψ
Luco and Westmann (1970)
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Layer
Effects
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Impedance Functions
ψ
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Boundary Element
Stehmeyer and Rizos, 2006
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B-Spline Impulse Response Approach
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z  Zeit then
K    2 MZ  p
Mz Kz  peit
Finite/Hybrid
Model

G*  G 1  2   2i 1  
2
2

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Dynamic p-y Curves
Tahghighi and Tonagi 2007
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Soil Properties

Shear Modulus, G and Damping Ratio, D

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Soil Type
Confining Stress
Void Ratio
Strain Level
Field: Cross-Hole, Down-Hole, Surface
Analysis of Seismic Waves SASW
Laboratory: Resonant Column, Torsional
Simple Shear, Bender Elements
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Oscilloscope
Crosshole Testing
ASTM D 4428
Pump
t
Shear Wave Velocity:
Vs = x/t
Downhole
Hammer
(Source)
Test
Depth
x
packer
Note: Verticality of casing
must be established by
slope inclinometers to correct
distances x with depth.
Slope
Inclinometer
PVC-cased
Borehole
Velocity
Transducer
(Geophone
Receiver)
Slope
Inclinometer
PVC-cased
Borehole
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Resonant Column Test
G, D for Different γ
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Torsional Shear Test
Schematic
Stress-Strain
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Hollow Cylinder RC-TOSS
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TOSS Test Results
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Steam Turbine-Generator
(Moreschi and Farzam, 2003)
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Machine Foundation Design Criteria

Deflection criteria: maintain turbine-generator
alignment during machine operating conditions

Dynamic criteria: ensure that no resonance
condition is encountered during machine
operating conditions
Jump to Resonance
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Strength criteria: reinforced concrete design
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STG Pedestal Structure
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Vibration Properties Evaluation
Identification of the foundation natural
frequencies for the dominant modes
 Frequency exclusion zones for the natural
frequencies of the foundation system and
individual structural members (±20%)
 Eigenvalue analysis: natural frequencies,
mode shapes, and mass participation
factors

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Finite Element Model
Structure and Base
Z
Y
X
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Low Frequency Modes
1st mode
6.5 Hz
95 % m.p.f.
2nd mode
7.2 Hz
76 % m.p.f
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High Frequency Modes
28th mode
46.3 Hz
0.3% m.p.f
42nd mode
64.6 Hz
0.03% m.p.f
Excitation frequency: 50-60 Hz
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Local Vibration Modes

Identification of natural
frequencies for individual
structural members

Quantification of changes
on vibration properties due
to foundation modifications
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ATST Telescope and FE Model
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Assumptions in FE analyses




Optics Lab mass/Instrument weight = 228 tons
Wind mean force = 75 N, RMS = 89 N
Ground base excitation PSD = 0.004 g2/hz
Concrete Pier


High Strength Concrete (E=3.11010 N/m2,
=0.15)
Soil Stiffness, k

Four different values using Arya & O’Neil’s
formula based on the site test data (Shear
modulus:30~75ksi, Poisson’s ratio:0.35~0.45)
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Frequency vs Soil Stiffness
Stiffness units = SI, frequency mode (hz)
Stiffness
Kx
Ky
Kz
Krx
Kry
Krz
MODE
1
2
3
4
5
6
min
1.19E+10
1.19E+10
1.48E+10
1.34E+12
1.34E+12
1.74E+12
6.3
6.4
9.4
9.4
10.4
11.2
1.83E+10
1.83E+10
2.45E+10
2.21E+12
2.21E+12
2.61E+12
2.48E+10
2.48E+10
3.41E+10
3.09E+12
3.09E+12
3.49E+12
max
3.12E+10
3.12E+10
4.38E+10
3.96E+12
3.96E+12
4.36E+12
7.0
7.1
9.7
10.3
11.9
13.0
7.4
7.5
9.9
11.1
12.6
13.6
7.5
7.7
10
11.8
13.3
13.7
m in+33.3% m in+66.6%
• Soil property range: Shear modulus (30~75ksi), Poisson’s ratio (0.35~0.45)
• Pier Footing: Diameter (23.3m)
• “min” for shear modulus of 30 ksi; “max” for 75 ksi
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Summary and Conclusions (Cho, 2005)
1.
2.
3.
4.
5.
6.
7.
8.
9.
High fidelity FE models were created
Relative mirror motions from zenith to horizon pointing: about 400 mm in
translation and 60 mrad in rotation.
Natural frequency changes by 2 hz as height changes by 10m.
Wind buffeting effects caused by dynamic portion (fluctuation) of wind
Modal responses sensitive to stiffness of bearings and drive disks
Soil characteristics were the dominant influences in modal
behavior of the telescopes.
Fundamental Frequency (for a lowest soil stiffness):
OSS=20.5hz; OSS+base=9.9hz; SS+base+Coude+soil=6.3hz
A seismic analysis was made with a sample PSD
ATST structure assembly is adequately designed:
1.
Capable of supporting the OSS
2.
Dynamically stiff enough to hold the optics stable
3.
Not significantly vulnerable to wind loadings
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Free-Field Analytical Solutions
ur
uz
 r 
 L V 
u z (r , ,0)  i  03  RV (a0 ) H 02  
 2  
 CR 
  M 0V 
2  r 
 

ur (r , ,0)  i 
R
(
a
)
H
V
0
1 
3

 2  
 CR 
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Trench
Isolation
Karlstrom and Bostrom 2007
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Chehab and Nagger 2003
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Celibi et al (in press)
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Thank-you

Questions?
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r -2
r -2 r -0.5
+
-
+
Rayleigh wave
Vertical
Horizontal
component component
+
Shear
wave
+
Relative
amplitude
-
r -1
+
Shear
window
r -1
+
r
Wave Type
Percentage of
Total Energy
Rayleigh
67
Shear
26
Compression
7
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Waves
Rayleigh, R
Surface
Shear,S
Secondary
Compression, P
Primary
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Machine Performance Chart
Performance Zones
A=No Faults, New
B=Minor Faults,
Good Condition
C = Faulty, Correct
In 10 Days To Save
$$
0.002
D = Failure Is Near,
Correct In 2 Days
E = Stop Now
450
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