vibrations of machine foundations

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Vibrations of

Machine Foundations

Richard P. Ray, Ph.D., P.E.Civil and Environmental Engineering

University of South Carolina

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Thanks To:Prof. Richard D. Woods, Notre Dame Univ.

Prof. F.E. Richart, Jr.

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Topics for Today

Fundamentals Modeling Properties Performance

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Foundation Movement

X

Z

Y

θ

ψ

φ

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How Does It Fail? 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)

Design Questions (1/4)

Fundamentals-Modeling-Properties-Design-Performance

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Velocity Requirements

Massarch (2004) "Mitigation of Traffic-Induced Ground Vibrations"


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What Are Relations Between Loads And Failure Quantities Loading -Machine (Periodic), Impluse, Natural Relations Between Load, Structure, Foundation,

Soil, Neighboring Structures Generate Model: Deterministic or Probabilistic


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How Do We Measure What Is Necessary? Full Scale Tests Prototype Tests Small Scale Tests (Centrifuge) Laboratory Tests (Specific Parameters) Numerical Simulation


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What Factor of Safety Do We Use? Does FOS Have Meaning What Happens After There Is Failure

Loss of Life Loss of Property Loss of Production

Purpose of Project, Design Life, Value


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r -2 r -2 r -0.5

r -1

r -1

r

Shear wave

Vertical component

Horizontal component

Shear window

Rayleigh wave

Relative amplitude+

+

+

+

- -

+

+

Wave Type Percentage of

Total Energy

Rayleigh 67

Shear 26

Compression 7

Waves


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Modeling Foundations Lumped Parameter (m,c,k) Block System

Parameters Constant, Layer, Special Impedance Functions

Function of Frequency (ω), Layers Boundary Elements (BEM)

Infinite Boundary, Interactions, Layers Finite Element/Hybrid (FEM, FEM-BEM)

Complex Geometry, Non-linear Soil


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Lumped Parameter)sin( tPP o

m

Gk

m

cν ρ

)sin(0 tPkzzczm

r


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SDOF

222

21

1

nn

D

static

dynamic

A

AMag


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Lumped Parameter System

Kx

Z

ψ

KzCz

Cx

Kψ

Cψ/2 Cψ/2

X

)sin(0 tPzkzczm zzz

mkcccD crcr 2

m

kn

mIψ


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Lumped Parameter Values

Mode Vertical Horizontal Rocking Torsion

Stiffness k

Mass Ratio m

Damping Ratio, D Fictitious

Mass

1

4Gr

2

8Gr)1(3

8 3

Gr

3

16 3Gr

5r

I

38

)2(

r

m

34

)1(

r

m

2/1ˆ425.0

m 2/1ˆ288.0

m2/1ˆ)ˆ1(

15.0

mm m̂21

50.0

m

mˆ

27.0m

mˆ

095.0

m

I x

ˆ24.0

m

I z

ˆ24.0

m̂

D=c/ccr G=Shear Modulus ν=Poisson's Ratio r=Radius ρ=Mass Density Iψ,Iθ=Mass Moment of Inertia

58

)1(3

r

I


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Mass Ratio


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Design Example 1VERTICAL 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

Shear Wave Velocity Vs = 680 ft/sec

Shear Modulus, G = 11 000 psi

Density, γ = 110 lb/ft3

Poisson's Ratio, ν = 0.33

DESIGN CRITERION:

Smooth Operation At Speed

Velocity <0.10 in/sec

Displacement < 0.002 in

Jump to Chart


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rGr

Q

k

QA

zzs

000114

)18857720(667.0

4

)1("002.0 00

'07.6"8.72 r

"002.0

2

10.166.0

ˆ

425.0

42.018.61104

9006467.0

4

)1(ˆ

33

staticzdynamicz

z

AA

DM

mD

g

g

W

rm

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

Jump to Figure


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Design Example - Table Top18'

34'

18'11'

Q0=400 lb

ψ

W=550 000 lb

Iψ=2.88 x 106 ft-lb-sec2

Soil Properties

Shear Wave Velocity Vs = 770 ft/sec

Shear Modulus, G = 14 000 psi

Density, γ = 110 lb/ft3

Poisson's Ratio, ν = 0.33

DESIGN CRITERION

0.20 in/sec Horizontal Motion at Machine Centerline

Ax = 0.0015 in. from combined rocking and sliding

Speed = 160 rpm

Slower speeds, Ax can be larger


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Horizontal Translation Only

inGr

Q

k

QAMag

mD

r

mmft

cdrtEquivanlen

xstaticxx

5002/1

3

100.32

80.1465.0

ˆ288.0

38.08

2ˆ96.13

341844

Rocking About Point "O"

.7200184006.509.0ˆ)ˆ1(

15.0

83.0)04.12(

2.32110

1088.2

8

)67.0(3

8

)1(3ˆ

/101088.2

1090.2/1090.2

33.02

04.12)14400014(8

2

8

/5.121200.123

91716

3

16

5

6

5

6

88

4

3

4

3

lbsftMMomentStaticMagmm

D

r

Im

secradI

kftlb

Grk

secradrpmftcd

rEquivalent

o

n


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.106.5)100.1(6.5

100.1)1218(10

50.0

10

50.0

109.2

)67.0(37200

44

46

68

inResonanceAt

inhAMotionHorizontal

radk

MDeflectionAngularStatic

sxs

os


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Impedance Methods

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

)sin()cos( titPePP oti

o

SOILSTATIC

z

zz D

KCikKCiK

A

RS

2

)(

Radiation DampingSoil Damping

Jump Wave

Sz


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Impedance Functions

Luco and Westmann (1970)

sV

r

Gra

0


ψ

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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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Finite/Hybrid Model

tie pzKzM

22 1221* iGG

pZMK

Zz

2

thene ti


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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

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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Crosshole TestingOscilloscope

PVC-cased Borehole

PVC-cased Borehole

DownholeHammer (Source) Velocity

Transducer (GeophoneReceiver)

t

x

Shear Wave Velocity:Vs = x/t

TestDepth

ASTM D 4428

Pump

packer

Note: Verticality of casingmust be established by

slope inclinometers to correctdistances x with depth.

SlopeInclinometer

SlopeInclinometer

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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

Strength criteria: reinforced concrete design


Jump to Resonance

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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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XYZ

XYZ

Finite Element Model Structure and Base


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Low Frequency Modes

1st mode6.5 Hz

95 % m.p.f.

2nd mode7.2 Hz

76 % m.p.f


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High Frequency Modes

28th mode46.3 Hz

0.3% m.p.f

42nd mode64.6 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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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=3.11010 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)

Assumptions in FE analyses


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• 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

Frequency vs Soil Stiffness

Stiffness min min+33.3% min+66.6% maxKx 1.19E+10 1.83E+10 2.48E+10 3.12E+10Ky 1.19E+10 1.83E+10 2.48E+10 3.12E+10Kz 1.48E+10 2.45E+10 3.41E+10 4.38E+10Krx 1.34E+12 2.21E+12 3.09E+12 3.96E+12Kry 1.34E+12 2.21E+12 3.09E+12 3.96E+12Krz 1.74E+12 2.61E+12 3.49E+12 4.36E+12

6.3 7.0 7.4 7.56.4 7.1 7.5 7.79.4 9.7 9.9 109.4 10.3 11.1 11.810.4 11.9 12.6 13.311.2 13.0 13.6 13.7

4

5

6

MODE1

2

3

Stiffness units = SI, frequency mode (hz)


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Summary and Conclusions (Cho, 2005)

1. High fidelity FE models were created2. Relative mirror motions from zenith to horizon pointing: about 400 m

in translation and 60 rad in rotation.3. Natural frequency changes by 2 hz as height changes by 10m.4. Wind buffeting effects caused by dynamic portion (fluctuation) of wind 5. Modal responses sensitive to stiffness of bearings and drive disks

6. Soil characteristics were the dominant influences in modal behavior of the telescopes.

7. Fundamental Frequency (for a lowest soil stiffness): OSS=20.5hz; OSS+base=9.9hz; SS+base+Coude+soil=6.3hz

8. A seismic analysis was made with a sample PSD9. ATST structure assembly is adequately designed:

1. Capable of supporting the OSS2. Dynamically stiff enough to hold the optics stable3. Not significantly vulnerable to wind loadings


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Free-Field Analytical Solutions

RVz C

rHaR

VLiru

2003

0 )(2

)0,,(

RVr C

rHaR

VMiru

2103

0 )(2

)0,,(

ur

uz


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Karlstrom and Bostrom 2007

Trench Isolation


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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

r -1

r -1

r

Shear wave

Vertical component

Horizontal component

Shear window

Rayleigh wave

Relative amplitude+

+

+

+

- -

+

+

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 ChartPerformance Zones

A=No Faults, New

B=Minor Faults, Good Condition

C = Faulty, Correct In 10 Days To Save $$

D = Failure Is Near, Correct In 2 Days

E = Stop Now

0.002

450

vibrations of machine foundations

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