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    Constructed Layered Systems: Measurements and Analysis
    Constructed Layered Systems: Measurements and Analysis
    Constructed Layered Systems: Measurements and Analysis
    Ebook273 pages1 hour

    Constructed Layered Systems: Measurements and Analysis

    By W. H. Cogill

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    About this ebook

    The study of constructed layered systems is based on the results of the measurements of the velocity of waves on the free surface of a system. The velocity varies with the frequency. The manner of the variation can be interpreted mathematically to estimate the depths of the interfaces and the elastic properties of the component media of the system.

    The waves can be generated by the generated continuous oscillation of a point, usually at the free surface of the system. Alternatively, impact loads can be applied.

    Horizontally-polarized shear waves (SH waves) yield the most precise interpreted measurement of the depths of the interfaces. However waves of the Rayleigh type are easier to generate, needing only an impact load on the free surface of the system.

    Theoretical results obtained from elementary structures are given. These are followed by Fortran programs with examples. The examples are intended to be used as guide to interpret the results of the measurements on the free surface of an actual structure.

    An actual structure, such as an airport runway, contains weaknesses which may lead to catastrophic failure of the runway. The compaction of the granular materials composing the runway affects the observed properties of the system. Theoretical development, and practical results of measurements, are given to indicate methods of correcting the weaknesses of the runway.

    LanguageEnglish
    PublisherXlibris AU
    Release dateJun 15, 2012
    ISBN9781477112847
    Constructed Layered Systems: Measurements and Analysis

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      Book preview

      Constructed Layered Systems - W. H. Cogill

      Copyright © 2012 by W. H. Cogill.

      Library of Congress Control Number:        2012908805

      ISBN:                  Hardcover                       978-1-4771-1283-0

      ISBN:                  Softcover                        978-1-4771-1282-3

      ISBN:                  Ebook                             978-1-4771-1284-7

      All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

      To order additional copies of this book, contact:

      Xlibris Corporation

      1-800-618-969

      www.Xlibris.com.au

      [email protected]

      501215

      Contents

      Chapter 1              SH-waves

      Chapter 2             Single-layered inverse

      Chapter 3             Inverse Problem

      Chapter 4             Results of Measurements

      Chapter 5             Computation

      Chapter 6             Comments

      Chapter 7             Program listings

      Chapter 8             Experimental Results

      Chapter 9             Computations

      Chapter 10           Application of Vand’s formula

      Bibliography

      END NOTES

      Chapter 1

      SH-waves

      1.1 Introduction

      Horizontally-polarized shear waves (SH-waves) may be utilized for a purpose similar to that of waves of the Rayleigh type [3,10]. In this chapter, we consider the response to be expected from a system composed of two layers each having a constant thickness, and resting on an underlying medium which is semi-infinite in extent.

      1.2 A system consisting of two layers overlying a semi-infinite medium

      SH-fund

      The system is defined in terms of cylindrical co-ordinates (r, θ, z) with the origin at the free surface and z measured positively downwards. The particle displacements are u and v in the r and θ directions respectively. The stresses are denoted by subscripts, for example, (pθθ)m, m = 1, 2, 3. The suffix 1 denotes quantities in the upper layer, 2 and 3 those in the lower layer and in the semi-infinite medium respectively.

      From Nakano [18], the following expressions for the peak values of the shear stresses and displacements are obtained:

      (a) In the surface layer,

      Image14825.EPS

      (1.1)

      Image14831.EPS

      (1.2)

      Image14837.EPS

      (1.3)

      Image14843.EPS

      (1.4)

      (b) In the second layer,

      Image14849.EPS

      (1.5)

      Image14855.EPS

      (1.6)

      Image14861.EPS

      (1.7)

      Image14867.EPS

      (1.8)

      (c) In the semi-infinite medium, we take s3 to be positive in order that the displacements may vanish as the value of z tends to infinity,

      Image14873.EPS

      (1.9)

      Image14879.EPS

      (1.10)

      Image14885.EPS

      (1.11)

      Image14891.EPS

      (1.12)

      In writing equations (1.9) to (1.12) one boundary condition, that of zero displacement at infinite depth, has already been taken into account. The arbitrary constants A1, A′1, A2, A′2, and A3 can be eliminated using the conditions at the interfaces within the system. The boundary condition at the free surface, namely that the shear stresses are zero in the r, θ plane, is not required.

      At the first interface, the stresses and displacements are continuous[7], i.e. when z = f

      u1 = u2, v1 = v2, (1.13)

      (pzr)1 = (pzr)2, (pθz)1 = (pθz)2

      At the second interface, the stresses and displacements are continuous, i.e. when z = f + g

      u2 = u3, v2 = v3, (1.14)

      (pzr)2 = (pzr)3, (pθz)2 = (pθz)3

      Owing to similarities between equations (1.1) to (1.4) and (1.5) to (1.8), the conditions relating to displacement and shear stress lead to only one equation.

      The use of the boundary conditions (1.13), with equations (1.1) to (1.4) and (1.5) to (1.8), leads to the following equations:

      Image14897.EPS

      (1.15)

      and

      Image14903.EPS

      (1.16)

      The use of the boundary conditions (1.14) with equations (1.5) to (1.8) and (1.9) to (1.12), leads to the following equations:

      Image14909.EPS

      (1.17)

      and

      Image14915.EPS

      (1.18)

      As one of the arbitrary constants (A3) is known to be non-zero, A1, A′1, A2, A′2, and A3 can be eliminated from equations (1.15), (1.17) and (1.18), and the following wave velocity equation is obtained [21]:-

      (μ3s3 cosh s2g + μ2s2 sinh s2g) μ2s2 cosh s1f

      + (μ3s3 sinh s2g + μ2s2 cosh s2g) μ1s1 sinh s1f = 0 (1.19)

      Several forms of equation (1.19) are possible. These depend on whether the displacements are considered to vary exponentially or in an oscillatory manner with the depth. From (1.19), there is no possibility of oscillatory variation in all three media, as the equation yields no solution if s1, s2, and s3 are all considered imaginary.¹

      1.3 Forms of Equation (1.19)

      Four forms of equation (1.19) are considered.

      Cases14

      1.3.1 Case 1

      Write Image14977.EPS . From equation (1.19) there is obtained

      Image14983.EPS

      Put Image14989.EPS , then

      Image14995.EPS

      .

      On writing Image15001.EPS , s2, and s3 in full and re-arranging, there is obtained

      Image15007.EPS

      (1.22)

      1.3.2 Case 2

      Write Image15013.EPS . From equation (1.19) there is obtained

      Image15019.EPS

      (1.23)

      Put Image15025.EPS , then

      Image15031.EPS

      .

      On writing s1, Image15037.EPS and s3 in full and re-arranging, there is obtained

      Image15043.EPS

      (1.24)

      1.3.3 Case 3

      Write Image15049.EPS , Image15055.EPS . From equation (1.19) there is obtained

      Image15061.EPS

      . (1.25)

      Put Image15067.EPS , then

      Image15073.EPS

      .

      On writing Image15079.EPS , Image15085.EPS , and s3 in full and re-arranging, there is obtained

      Image15091.EPS

      (1.26)

      1.3.4 Case 4

      From equation (1.19) there is obtained

      Image15097.EPS

      . (1.27)

      Put Image15103.EPS , then

      Image15109.EPS

      .

      On writing s1, s2, and s3 in full and re-arranging, there is obtained

      Image15115.EPS

      (1.28)

      1.4 Matrix Methods

      Most systems which are of practical importance are more complex that those considered in the first part of this chapter. They contain layers of materials having a wide range of stiffnesses. The layers may be interspersed in any order of stiffness. The deepest layer of which the system is composed is usually regarded as extending to infinite depth and is represented by a semi-infinite medium. A method is required of calculating the phase velocities of SH waves at the free surfaces of these structures. It is expected that the phase velocities will vary with the frequency. The phase velocities will be complex quantities if leaking modes are considered (Su and Dorman [25]).

      The method proposed by Thomson [26] will be followed, and use will made of the notation employed by Thrower [27]. The component materials are considered as purely elastic, and leaking modes are not considered. The vector representing the shear stress pθz and the displacement v in the m-th medium from the free surface can be written

      Image15121.EPS (1.29)

      The peak value of the shear stress can be written (Nakano [18])

      Image15127.EPS

      (1.30)

      Image15133.EPS

      , (1.31)

      where S = A′ + A and S′ = A′-A, and the peak value of the displacement can be written

      Image15139.EPS

      (1.32)

      Image15145.EPS

      . (1.33)

      Here C′ is some circular function, the exact form of which is not important as it cancels when the boundary conditions are applied. As the trajectory of the particle motion is horizontal and purely tangential to the direction of propagation, only one shear stress and one displacement need be considered. This suffices to determine the stresses and displacements throughout the system.

      In matrix notation

      Image15151.EPS (1.34)

      where

      Image15157.EPS and (1.35)

      Image15163.EPS

      (1.36)

      where the suffix m denotes that the quantities are those which apply to the m-th medium below the free surface of the structure. Placing the origin of the coordinates at the (m-1)-th interface, the expression for the stress-displacement vector Image15169.EPS , within the m-th layer at the (m-1)-th interface is

      Image15175.EPS

      (1.37)

      where Image15181.EPS is derived from Image15187.EPS by putting z = 0. The stress-displacement vector Image15193.EPS within the m-th layer at the m-th interface is

      Image15199.EPS

      (1.38)

      where Image15205.EPS is derived from Image15211.EPS by putting z = Hm, the thickness of the m-th layer. Equation (1.37) can be written

      Image15217.EPS

      (1.39)

      The vectors Image15223.EPS and Image15229.EPS , representing the stresses and displacements at the top and bottom of the m-th layer, are therefore related by

      Image15235.EPS

      (1.40)

      It is assumed that there is

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