Wave Travel and Attenuation and Machine Foundations Richard P. Ray, Ph.D., P.E. Civil and Environmental Engineering University of South Carolina

Topics for Today Waves in Elastic Media Waves in the Earth Surface Excitations Machine Foundations

Waves Rayleigh, R Surface (2-D) Shear,S Secondary (1-D) Compression, P Primary (1-D)

Resonant Column - MOC - Wavelets Discrete Properties

Resonant Column - MOC - Wavelets Rock Motion Soil 1: G 1,ρ 1,μ 1 Soil j: G j,ρ j,μ j Soil m: G m,ρ m,μ m Surface Block Mass Horizontal Polarization Vertical Propagation Computational Reaches Nodes Δz n-1 Δz 1 Δz i τi,Viτi,Vi A B P t=

Resonant Column - MOC - Wavelets  t C + characteristic: C - characteristic:  =shearing stress; V=particle velocity. S =phase (shear wave) velocity;  =mass density; t=time; B5B5 C1C1  z3z3  z2z2  z1z1  z4z4 A1A1 A2A2 A3A3 B2B2 B3B3 B4B4 C2C2 C3C3 C4C4 A4A4 C5C5 P1P1 P2P2 P3P3 P4P4 P5P5 C-C- C-C- C-C- C-C- C+C+ C+C+ C+C+ C+C+

Resonant Column - MOC - Wavelets S time P B A B2 A2 CC2C3 B3 A3 C + C -  t  z R time R space S space Nonlinear Interpolation

Resonant Column - MOC - Wavelets ΔzΔz 30 31

Resonant Column - MOC - Wavelets

Cumulative Hysteretic Energy Time (sec) Reach Number Strain 400Hyst 400

A1A1 A2A2 A3A3 A4A4 Wavelets

Resonant Column - MOC - Wavelets

Profile View

MEMS Accelerometer

Data Acquisition

Resonant Column - MOC - Wavelets Wavelets

Resonant Column - MOC - Wavelets Wavelets

Resonant Column - MOC - Wavelets By varying the wavelet scale s and translating along the localized time index n, one can construct a picture showing both the amplitude of any features versus the scale and how this amplitude varies with time. Wavelet Scale Localized Time Index Fourier Transform Wavelet via Fourier Transform

Resonant Column - MOC - Wavelets

Wavelets

r -2 r -0.5 r -1 r Shear wave Vertical component Horizontal component Shear window Rayleigh wave Relative amplitude Wave TypePercentage of Total Energy Rayleigh67 Shear26 Compression7 Waves Fundamentals-Modeling-Properties-Performance

Free-Field Analytical Solutions urur uzuz Fundamentals-Modeling-Properties-Performance

Free-Field Analytical Solutions urur uzuz Fundamentals-Modeling-Properties-Performance

Karlstrom and Bostrom 2007 Trench Isolation Fundamentals-Modeling-Properties-Performance

Chehab and Nagger 2003 Fundamentals-Modeling-Properties-Performance

Celibi et al (in press)

ATST Telescope and FE Model Fundamentals-Modeling-Properties-Performance

Summary and Conclusions (Cho, 2005) 1.High fidelity FE models were created 2.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 (dynamic) 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 PSD 9.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 Fundamentals-Modeling-Properties-Performance

Foundation Movement X Z Y θ ψ φ Fundamentals-Modeling-Properties-Performance

Design Questions (1/4) How Does It Fail? Static Settlement Dynamic Motion Too Large (0.02 mm) Settlements Caused By Dynamic Motion Liquefaction What Are Maximum Values of Failure? (Acceleration, Velocity, Displacement) Fundamentals-Modeling-Properties-Design-Performance

Velocity Requirements Massarch (2004) "Mitigation of Traffic-Induced Ground Vibrations" Fundamentals-Modeling-Properties-Performance 0,40

Fundamentals-Modeling-Properties-Performance

Design Questions (2/4) What Are Relations Between Loads And Failure Quantities? Loads -Harmonic, Periodic, Random Load→ Structure → Foundation → Soil → Neighboring Structures Model: Deterministic or Probabilistic Fundamentals-Modeling-Properties-Performance

Design Questions (3/4) How Do We Measure What Is Necessary? Full Scale Tests Prototype Tests Small Scale Tests (Centrifuge) Laboratory Tests (Specific Parameters) Computer Model Fundamentals-Modeling-Properties-Performance

Design Questions (4/4) 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 Fundamentals-Modeling-Properties-Performance

r -2 r -0.5 r -1 r Shear wave Vertical component Horizontal component Shear window Rayleigh wave Relative amplitude Wave TypePercentage of Total Energy Rayleigh67 Shear26 Compression7 Waves Fundamentals-Modeling-Properties-Performance

r -2 r -0.5 r -1 r Shear wave Vertical component Horizontal component Shear window Rayleigh wave Relative amplitude Wave TypePercentage of Total Energy Rayleigh67 Shear26 Compression7 Waves Fundamentals-Modeling-Properties-Performance

Modeling Foundations Lumped Parameter (m,c,k) Block System Parameters Constant, Layers, 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 Fundamentals-Modeling-Properties-Performance

Lumped Parameter m G k m c ν ρ r Fundamentals-Modeling-Properties-Performance

Single Degree of Freedom k m c z

c=0…Undamped c=2mω…Critically Damped c<2mω…Underdamped

Single Degree of Freedom z(0) t

Single Degree of Freedom z(0) t

Single Degree of Freedom See Chart

Single Degree of Freedom k m c

SDOF Transient and Steady-State

Fundamentals-Modeling-Properties-Performance

Lumped Parameter System KxKx Z ψ KzKz CzCz CxCx KψKψ C ψ /2 X m IψIψ Fundamentals-Modeling-Properties-Performance

Lumped Parameter Values Mode Vertical z Horizontal x Rocking ψ Torsion θ Stiffness k Mass Ratio m Damping Ratio, D D=c/c cr G=Shear Modulus ν=Poisson's Ratio r=Radius ρ=Mass Density I ψ,I θ =Mass Moment of Inertia Fundamentals-Modeling-Properties-Performance

Design Example 1 VERTICAL COMPRESSOR Unbalanced Forces Vertical = 45 kN Horzontal Primary = 0,5 kN Operating Speed = 450 rpm Wt Machine + Motor = kg Soil Properties Shear Wave Velocity V s = 250 m/sec Density, ρ = 1600 kg/m 3 Shear Modulus, G = 1,0e8 Pa Poisson's Ratio, ν = 0,33 DESIGN CRITERION: Smooth Operation At Speed Velocity <0,10 in/sec Displacement < 0,002 in <0,05mm Jump to Chart Fundamentals-Modeling-Properties-Performance

Try a 3 x 2,5 x 1 foundation block, r = 1,55 m Mass = kg Total Mass = = kg Jump to Figure Fundamentals-Modeling-Properties-Performance

Design Example - Table Top 5m 10m 5m 4m Q 0 =1800 N ψ m= kg I ψ =1,0 x 10 7 N-m-sec 2 Soil Properties Shear Wave Velocity V s = 200 m/sec Shear Modulus, G = 6,80x10 7 Pa Density, γ = 1700 kg/m 3 Poisson's Ratio, ν = 0,33 DESIGN CRITERION 5.0 mm/sec Horizontal Motion at Machine Centerline X = 0,04 mm from combined rocking and sliding Speed = 160 rpm Slower speeds, X can be larger Fundamentals-Modeling-Properties-Performance X

Horizontal Translation Only Rocking About Point "O" Fundamentals-Modeling-Properties-Performance Ax = 40x10 -3 mm

Fundamentals-Modeling-Properties-Performance ψ X X = 40x10 -3 mm

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 Fundamentals-Modeling-Properties-Performance

Impedance Functions Radiation Damping Soil Damping Jump Wave SzSz Fundamentals-Modeling-Properties-Performance