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Foundation Vibration Analysis Using Simple Physical Models

Book Description

This book provides simple physical models to represent the unbounded soil in time and frequency domain analysis. They do not supplant the more generally applicable rigorous methods, but rather supplement them. The physical models used consists of the following representations: cones based one-dimensional rod theory; lumped-parameter models with frequency-independent springs, dashpots, and masses; and prescribed wave patterns in the horizontal plane. The physical models thus offer a strength-of-materials approach to foundation dynamics.

Table of Contents

  1. Copyright
  2. Foreword
  3. Preface
  4. List of Tables for Lumped-Parameter Models
    1. Homogeneous Soil Halfspace
      1. Surface Foundation
      2. Embedded Foundation
    2. Soil Layer on Rigid Rock
      1. Surface Foundation
      2. Embedded Foundation
  5. 1. Introduction
    1. 1.1. Statement of Problem
    2. 1.2. Design of Machine Foundation
    3. 1.3. Effects of Soil-Structure Interaction
    4. 1.4. Classification of Analysis Methods
      1. 1.4.1. Direct Method and Substructure Method
      2. 1.4.2. Frequency Domain and Time Domain
      3. 1.4.3. Rigorous Boundary-Element Method and Approximate Simple Physical Model
    5. 1.5. Physical Modeling of Unbounded Soil
      1. 1.5.1. Applications
      2. 1.5.2. Overview
      3. 1.5.3. Examples
      4. 1.5.4. Requirements
    6. 1.6. Organization of Text
    7. Summary
  6. 2. Foundation on Surface of Homogeneous Soil Halfspace
    1. 2.1. Construction of Cone Model
      1. 2.1.1. Elastic Properties and Wave-Propagation Velocities
      2. 2.1.2. Aspect Ratio
      3. 2.1.3. High-Frequency Behavior
      4. 2.1.4. Nearly Incompressible Soil
        1. Wave velocity
        2. Trapped mass
    2. 2.2. Translational Cone
      1. 2.2.1. Stiffness Formulation
        1. Interaction force-displacement relationship
        2. Dynamic-stiffness coefficient
      2. 2.2.2. Flexibility Formulation
        1. Displacement-interaction force relationship
        2. Unit-impulse response function
        3. Recursive evaluation
        4. Nearly incompressible soil
        5. Dynamic-flexibility coefficient
      3. 2.2.3. Example for Hand Calculation
      4. 2.2.4. Vibration of Machine Foundation
        1. Crank mechanism of reciprocating machine
        2. Dynamic load of single-cylinder compressor
        3. Dynamic load of weaver’s loom
        4. Design criterion
        5. Rigid foundation block
        6. Soil
        7. Dynamic system
        8. Stiffness formulation in time domain
        9. Flexibility formulation in time domain
        10. Stiffness formulation for harmonic loading
        11. Alternative descriptions for harmonic motion
    3. 2.3. Rotational Cone
      1. 2.3.1. Stiffness Formulation
        1. Interaction moment-rotation relationship
        2. Recursive evaluation
        3. Discrete-element model
        4. Alternative general-purpose computer program implementation
        5. Dynamic-stiffness coefficient
      2. 2.3.2. Flexibility Formulation
        1. Rotation-interaction moment relationship
        2. Unit-impulse response function
        3. Recursive evaluation
        4. Dynamic-flexibility coefficient
      3. 2.3.3. Example for Hand Calculation
      4. 2.3.4. Vibration of Machine Foundation
        1. Dynamic load of three-cylinder compressor with cranks at 120°
        2. Design criterion
        3. Rigid foundation block
        4. Soil
        5. Uncoupled dynamic system with rocking motion only
        6. Uncoupled stiffness formulation in time domain
        7. Uncoupled stiffness formulation for harmonic loading
        8. Dynamic system with coupling of rocking and horizontal motions
        9. Coupled stiffness formulation in time domain
        10. Coupled stiffness formulation for harmonic loading
        11. Dynamic system of torsional motion
    4. 2.4. Material Damping
      1. 2.4.1. Correspondence Principle Applied for Harmonic Motion
      2. 2.4.2. Correspondence Principle Applied to Discrete-Element Model
      3. 2.4.3. Comparison of Models with Linear-Hysteretic Damping and Voigt Viscoelasticity
      4. 2.4.4. Frictional Material Damping
        1. Frictional discrete-element model
        2. Harmonic motion of translational cone
        3. Harmonic motion of rotational cone
      5. 2.4.5. Example of Seismic Analysis
    5. 2.5. Why a Cone Model Can Represent the Elastic Halfspace
      1. 2.5.1. Object
      2. 2.5.2. Strength-of-Materials Versus Rigorous Theory of Halfspace
        1. Static case
        2. Dynamic case (radiation criterion)
        3. Conical nature of rigorous solution
        4. Equation of motion for unified cone model
        5. Wave velocity and trapped mass for vertical and rocking motions in case of nearly incompressible soil
      3. 2.5.3. Neglect of Portion of Halfspace Outside Cone
        1. Static case
        2. Dynamic case
        3. Estimation of cone’s aspect ratio
      4. 2.5.4. Inability of Cone to Represent Rayleigh Wave
    6. 2.6. Simple Lumped-Parameter Models
      1. 2.6.1. Standard Lumped-Parameter Model
      2. 2.6.2. Fundamental Lumped-Parameter Model
      3. 2.6.3. Torsional Motion of Disk with Polar Mass Moment of Inertia on Halfspace
    7. 2.7. Translational and Rocking Wedges
      1. 2.7.1. Dynamic-Stiffness Coefficient
      2. 2.7.2. Aspect Ratio
      3. 2.7.3. Comparison of Dynamic-Stiffness Coefficients for Wedge with Exact Solution
    8. 2.8. Insight on Two-Dimensional Versus Three-Dimensional Foundation Modeling
      1. 2.8.1. Object
      2. 2.8.2. Equivalent Slice of Two-Dimensional Strip Foundation with Same Material Properties
      3. 2.8.3. Alternative Equivalent Slice of Two-dimensional Strip Foundation Adjusting Impedance
      4. 2.8.4. Dynamic-Stiffness Coefficients of Disk and Equivalent Slices of Strips
      5. 2.8.5. Decay of Waves
      6. 2.8.6. Row of Point Loads
      7. 2.8.7. Square Versus Slender Rectangular Foundations
    9. Summary
  7. 3. Foundation on Surface of Soil Layer on Rigid Rock
    1. 3.1. Construction of Unfolded Layered Cone Model
      1. 3.1.1. Layered Translational Cone
      2. 3.1.2. Layered Rotational Cone
    2. 3.2. Flexibility Formulation
      1. 3.2.1. Unit-Impulse Response Function
      2. 3.2.2. Static Case
    3. 3.3. Stiffness Formulation
      1. 3.3.1. Pseudo-Echo Constants for Rotation
      2. 3.3.2. Stiffness Echo Constants
      3. 3.3.3. Resonant Case for Translation
      4. 3.3.4. Nearly Incompressible Soil
      5. 3.3.5. Dynamic-Stiffness Coefficient
        1. Layered translational cone
        2. Layered rotational cone
        3. Example
        4. High-frequency behavior
    4. 3.4. Alternative Order of Realization
    5. 3.5. Illustrative Example
    6. 3.6. Soil Layer on Flexible Rock Halfspace
      1. 3.6.1. Reflection Coefficient for Translational Cone
      2. 3.6.2. Reflection Coefficient for Rotational Cone
      3. 3.6.3. Flexibility and Stiffness Formulations
        1. Translational cone
        2. Rotational cone
      4. 3.6.4. Accuracy Evaluation
        1. Static-stiffness coefficient
        2. Dynamic-stiffness coefficient
        3. Transfer function of layer on flexible halfspace to homogeneous half-space
    7. 3.7. Basic Lumped-Parameter Model
      1. 3.7.1. Introductory Example
      2. 3.7.2. Achieved Accuracy
      3. 3.7.3. Coefficients of Springs, Dashpots, and Mass
      4. 3.7.4. Rigid Block on Layer and on Halfspace with Partial Uplift
    8. 3.8. Cutoff Frequency
      1. 3.8.1. General Considerations
      2. 3.8.2. Practical Significance of Cutoff Frequency
      3. 3.8.3. Bar on Elastic Foundation
      4. 3.8.4. Criterion for Existence of Cutoff Frequency
    9. Summary
  8. 4. Embedded Foundation and Pile Foundation
    1. 4.1. Embedded Disk with Double-Cone Model and Its Green’s Function
      1. 4.1.1. One-Sided Cone
      2. 4.1.2. Double Cone
        1. Construction and static-stiffness coefficient
        2. Green’s function
      3. 4.1.3. Embedded Disk with Anti-Symmetric and Symmetric Mirror Disks To Model Free Surface and Fixed Boundary of Halfspace
      4. 4.1.4. Embedded Disk with Anti-Symmetric and Symmetric Mirror Disks To Model Free Surface and Fixed Boundary of Layer
      5. 4.1.5. Embedded Disk with Mirror Disk to Model Interface with Halfspace
    2. 4.2. Matrix Formulation for Harmonic Loading of Foundation Embedded in Halfspace
      1. 4.2.1. Dynamic-Flexibility Matrix of Free Field
      2. 4.2.2. Dynamic-Stiffness Matrix
      3. 4.2.3. Example
    3. 4.3. Matrix Formulation in Time Domain of Foundation Embedded in Halfspace
      1. 4.3.1. Dynamic-Flexibility Matrix of Free Field
        1. Recursive evaluation
        2. Time-discretized displacement-interaction force relationship
      2. 4.3.2. Interaction Force-Displacement Relationship
        1. Time-integration scheme
        2. Alternative modeling of excavated soil
        3. Nearly incompressible soil
      3. 4.3.3. Example
        1. Harmonic loading
        2. Transient loading
    4. 4.4. Foundation Embedded in Layer
    5. 4.5. Limitation of Single Cone Model
    6. 4.6. Fundamental Lumped-Parameter Model for Foundation Embedded in Halfspace
      1. 4.6.1. Coupled Horizontal and Rocking Motions
      2. 4.6.2. Coefficients of Spring, Dashpots, and Mass
      3. 4.6.3. Hammer Foundation with Partial Uplift of Anvil
    7. 4.7. Basic Lumped-Parameter Model for Foundation Embedded in Layer
      1. 4.7.1. Coupled Horizontal and Rocking Motions
      2. 4.7.2. Achieved Accuracy
      3. 4.7.3. Coefficients of Springs, Dashpots, and Mass
      4. 4.7.4. Hammer Foundation with Partial Uplift of Anvil
    8. 4.8. Single Pile
      1. 4.8.1. Analogy to Embedded Foundation
      2. 4.8.2. Matrix Formulation for Harmonic Loading of Pile Embedded in Halfspace
        1. Dynamic-flexibility matrix of free field
        2. Dynamic-stiffness matrix
      3. 4.8.3. Example of Single Floating Pile
    9. 4.9. Pile Group
      1. 4.9.1. Group Behavior
      2. 4.9.2. Dynamic-Interaction Factor
      3. 4.9.3. Matrix Formulation for Dynamic-Stiffness Matrix
      4. 4.9.4. Example of Floating 3 × 3 Pile Group
    10. Summary
  9. 5. Simple Vertical Dynamic Green’s Function
    1. 5.1. Green’s Function of Point Load on Surface of Halfspace
      1. 5.1.1. Static Case
      2. 5.1.2. Dynamic Case
    2. 5.2. Solutions Derived via Point-Load Green’s Function
      1. 5.2.1. Ring of Point Loads
      2. 5.2.2. Rigid Disk in Vertical Motion
    3. 5.3. Approximate Green’s Function of Disk
      1. 5.3.1. Illustrative Example
      2. 5.3.2. Summary of Equations
        1. Amplitude-reduction factor
        2. Phase angle
    4. 5.4. Matrix Formulation of Surface Foundation with Arbitrary Shape Modeled with Subdisks
    5. 5.5. Example of Square Surface Foundation
    6. 5.6. Through-Soil Coupling of Surface Foundations
    7. Summary
  10. 6. Seismic Excitation
    1. 6.1. Basic Equation of Motion of Substructure Method
      1. 6.1.1. Formulation in Total Displacements for Harmonic Excitation
      2. 6.1.2. Formulation in Total Displacements in Time Domain
      3. 6.1.3. Kinematic and Inertial Interactions
    2. 6.2. Free-Field Response of Site
    3. 6.3. Effective Foundation Input Motion
      1. 6.3.1. Vertical and Rocking Components of Rigid Surface Foundation Modeled with Subdisks
      2. 6.3.2. Rigid Surface Foundation Subjected to Horizontally Propagating Waves
        1. Horizontal component coinciding with direction of propagation
        2. Horizontal component perpendicular to direction of propagation
        3. Vertical component
        4. Time-domain analysis
      3. 6.3.3. Embedded Foundation Modeled with Stack of Disks
        1. Analysis for harmonic loading
        2. Time-domain analysis
        3. Example
    4. 6.4. Surface Foundation Modeled with Cones and Unfolded Cones
      1. 6.4.1. Homogeneous Soil Halfspace
      2. 6.4.2. Soil Layer on Rigid Rock
    5. 6.5. Foundation Modeled with Lumped-Parameter Model
    6. 6.6. Basic Equation of Motion of Direct Method
    7. Summary
  11. 7. Dynamic Soil-Structure Interaction
    1. 7.1. Coupled Structure-Soil System for Horizontal and Rocking Motions
      1. 7.1.1. Simplest Dynamic Model
      2. 7.1.2. Equations of Motion
      3. 7.1.3. Effect of Interaction with Soil on Response of Structure
        1. Dimensionless parameters
        2. Structural response
    2. 7.2. Equivalent One-Degree-of-Freedom System
      1. 7.2.1. Material Damping of Soil
      2. 7.2.2. Equivalent Natural Frequency, Damping Ratio and Input Motion
      3. 7.2.3. Parametric Study
    3. 7.3. Other Equivalent Systems
      1. 7.3.1. Dynamic Model for Vertical Motion
      2. 7.3.2. Redundant Coupled Structure-Soil System for Horizontal and Rocking Motions
      3. 7.3.3. Coupled Rigid Block-Soil System
    4. Summary
  12. A. Interaction Force-Displacement Relationship and Green’s Function of Cone Model
    1. A1. Translational Cone
      1. A1.1. Equation of Motion
      2. A1.2. Interaction Force-Displacement Relationship
      3. A1.3. Displacement-Interaction Force Relationship
      4. A1.4. Green’s Function
    2. A2. Rotational Cone
      1. A2.1. Equation of Motion
      2. A2.2. Interaction Moment-Rotation Relationship
      3. A2.3. Rotation-Interaction Moment Relationship
      4. A2.4. Green’s Function
    3. A3. Generalized Wave Pattern of Unfolded Layered Cone
  13. B. Consistent Lumped-Parameter Model
    1. B1. Partial-Fraction Expansion of Dynamic Stiffness for Harmonic Loading
      1. B1.1. Dynamic-Stiffness Coefficient
      2. B1.2. Polynomial-Fraction Approximation
      3. B1.3. Partial-Fraction Expansion
      4. B1.4. Stability Criterion and Rate of Energy Transmission
    2. B2. Discrete-Element Models for Partial-Fraction Expansion
      1. B2.1. Overview
      2. B2.2. Constant and Linear Terms
      3. B2.3. First-Order Term
      4. B2.4. Second-Order Term
    3. B3. Semi-Infinite Bar on Elastic Foundation
      1. B3.1. Dynamic-Stiffness Coefficient
      2. B3.2. Lumped-Parameter Model with Two Internal Degrees of Freedom
      3. B3.3. Lumped-Parameter Model with Three Internal Degrees of Freedom
      4. B3.4. Dynamic-Flexibility Coefficient in Time Domain
  14. C. Recursive Evaluation of Convolution Integral
    1. C1. Overview
    2. C2. First-Order Term
    3. C3. Second-Order Term
  15. D. Dynamic Stiffness of Foundation on or Embedded in Layered Soil Halfspace
    1. D1. Backbone Cone and Layered Halfspace
    2. D2. Dynamic-Stiffness Matrix of Layer
      1. D2.1. Translational Motion
      2. D2.2. Rotational Motion
      3. D2.3. Example
    3. D3. Cloning Property
    4. D4. Examples
  16. References