IntroductionE Mch 521, ACS 521Stress Waves in

Solid Media3 credit Graduate Course

- Penn State University
- Instructors
- Dr. Joseph L. Rose
- Dr. Cliff Lissenden
- Textbook
- Ultrasonic Guided Waves in Solid Media
- Joseph L. Rose Cambridge University Press

2014

Preface

- Text Ultrasonic Waves in Solid Media, 1999
- Nondestructive Evaluation
- Structural Health Monitoring
- Growth of Guided Waves 1985 to 2014
- Publications to 2000 and beyond
- University involvement (2 to 40)
- Commercialization piping example
- ASNT working on inspection certification, a new

method - ASME, DOT code requirements/developments

Table of Contents

- Nomenclature
- Preface
- Acknowledgments
- 1. Introduction
- 1.1 Background
- 1.2 A Comparison of Bulk versus Guided Waves
- 1.3 What Is an Ultrasonic Guided Wave?
- 1.4 The Difference Between Structural Health

Monitoring (SHM) and

Nondestructive Testing (NDT) - 1.5 Text Preview
- 1.6 Concluding Remarks
- 1.7 References

Table of Contents cont.

- 2. Dispersion Principles
- 2.1 Introduction
- 2.2 Waves in a Taut String
- 2.2.1 Governing Wave Equation
- 2.2.2 Solution by Separation of Variables
- 2.2.3 DAlemberts Solution
- 2.2.4 Initial Value Considerations
- 2.3 String on an Elastic Base
- 2.4 A Dispersive Wave Propagation Sample Problem
- 2.5 String on a Viscous Foundation
- 2.6 String on a Viscoelastic Foundation
- 2.7 Graphical Representations of a Dispersive

System - 2.8 Group Velocity Concepts
- 2.9 Exercises
- 2.10 References

Table of Contents cont.

- 3. Unbounded Isotropic and Anisotropic Media
- 3.1 Introduction
- 3.2 Isotropic Media
- 3.2.1 Equations of Motion
- 3.2.2 Dilatational and Distortional Waves
- 3.3 The Christoffel Equation for Anisotropic

Media - 3.3.1 Sample Problem
- 3.4 On Velocity, Wave, and Slowness Surfaces
- 3.5 Exercises
- 3.6 References

Table of Contents cont.

- 4. Reflection and Refraction
- 4.1 Introduction
- 4.2 Normal Beam Incidence Reflection Factor
- 4.3 Snells Law for Angle Beam Analysis
- 4.4 Critical Angles and Mode Conversion
- 4.5 Slowness Profiles for Refraction and

Critical Angle Analysis - 4.6 Exercises
- 4.7 References

Table of Contents cont.

- 5. Oblique Incidence
- 5.1 Introduction
- 5.2 Reflection and Refraction Factors
- 5.2.1 Solid-Solid Boundary Conditions
- 5.2.2 Solid-Liquid Boundary Conditions
- 5.2.3 Liquid-Solid Boundary Conditions
- 5.3 Moving Forward
- 5.4 Exercises
- 5.5 References

Table of Contents cont.

- 6. Waves in Plates
- 6.1 Introduction
- 6.2 The Free Plate Problem
- 6.2.1 Solution by the Method of Potentials
- 6.2.2 The Partial Wave Technique
- 6.3 Numerical Solution of the Rayleigh-Lamb

Frequency Equations - 6.4 Group Velocity
- 6.5 Wave Structure Analysis
- 6.6 Compressional and Flexural Waves
- 6.7 Miscellaneous Topics
- 6.7.1 Lamb Waves with Dominant Longitudinal

Displacements - 6.7.2 Zeros and Poles for a Fluid-Coupled

Elastic Layer - 6.7.3 Mode Cutoff Frequency
- 6.8 Exercises
- 6.9 References

Table of Contents cont.

- 7. Surface and Subsurface Waves
- 7.1 Background
- 7.2 Surface Waves
- 7.3 Generation and Reception of Surface Waves
- 7.4 Subsurface Longitudinal Waves
- 7.5 Exercises
- 7.6 References

Table of Contents cont.

- 8. Finite Element Method for Guided Wave

Mechanics - 8.1 Introduction
- 8.2 Overview of the Finite Element Method
- 8.2.1 Using the Finite Element Method to Solve

a Problem - 8.2.2 Quadratic Elements
- 8.2.3 Dynamic Problem
- 8.2.4 Error Control
- 8.3 FEM Applications for Guided Wave Analysis
- 8.3.1 2-D Surface Wave Generation in a Plate
- 8.3.2 Guided Wave Defect Detection in a

Two-Inch Steel Tube - 8.4 Summary
- 8.5 Exercises
- 8.6 References

Table of Contents cont.

- 9. The Semi-Analytical Finite Element Method
- 9.1 Introduction
- 9.2 SAFE Formulation for Plate Structures
- 9.3 Orthogonality-Based Mode Sorting
- 9.4 Group Velocity Dispersion Curves
- 9.5 Guided Wave Energy
- 9.5.1 Poynting Vector
- 9.5.2 Energy Velocity
- 9.5.3 Skew Effects in Anisotropic Plates
- 9.6 Solution Convergence of the SAFE Method
- 9.7 Free Guided Waves in an Eight-Layer

Quasi-Isotropic Plate - 9.8 SAFE Formulation for Cylindrical Structures
- 9.9 Summary
- 9.10 Exercises
- 9.11 References

Table of Contents cont.

- 10. Guided Waves in Hollow Cylinders
- 10.1 Introduction
- 10.2 Guided Waves Propagating in an Axial

Direction - 10.2.1 Analytic Calculation Approach
- 10.2.2 Excitation Conditions and Angular

Profiles - 10.2.3 Source Influence
- 10.3 Exercises
- 10.4 References

Table of Contents cont.

- 11. Circumferential Guided Waves
- 11.1 Development of the Governing Wave Equations

for Circumferential Waves - 11.1.1 Circumferential Shear Horizontal Waves

in a Single-Layer Annulus - 11.1.2 Circumferential Lamb Type Waves in a

Single-Layer Annulus - 11.2 Extension to Multiple-Layer Annuli
- 11.3 Numerical Solution of the Governing Wave

Equations for Circumferential Guided

Waves - 11.3.1 Numerical Results for CSH-Waves
- 11.3.2 Numerical Results for CLT-Waves
- 11.3.3 Computational Limitations of the

Analytical Formulation - 11.4 The Effects of Protective Coating on

Circumferential Wave Propagation in Pipe - 11.5 Exercises
- 11.6 References

Table of Contents cont.

- 12. Guided Waves in Layered Structures
- 12.1 Introduction
- 12.2 Interface Waves
- 12.2.1 Waves at a Solid-Solid Interface

Stoneley Wave - 12.2.2 Waves at a Solid-Liquid Interface

Scholte Wave - 12.3 Waves in a Layer on a Half Space
- 12.3.1 Rayleigh-Lamb Type Waves
- 12.3.2 Love Waves
- 12.4 Waves in Multiple Layers
- 12.4.1 The Global Matrix Method
- 12.4.2 The Transfer Matrix Method
- 12.4.3 Examples
- 12.5 Fluid Couples Elastic Layers
- 12.5.1 Ultrasonic Wave Reflection and

Transmission - 12.5.2 Leaky Guided Wave Modes
- 12.5.3 Nonspecular Reflection and Transmission
- 12.6 Exercises
- 12.7 References

Table of Contents cont.

- 13 . Source Influence on Guided Wave Excitation
- 13.1 Introduction
- 13.2 Integral Transform Method
- 13.2.1 A Shear Loading Example
- 13.3 Normal Mode Expansion Method
- 13.3.1 Normal Mode Expansion in Harmonic

Loading - 13.3.2 Transient Loading Source Influence
- 13.4 Exercises
- 13.5 References

Table of Contents cont.

- 14. Horizontal Shear
- 14.1 Introduction
- 14.2 Dispersion Curves
- 14.3 Phase Velocities and Cutoff Frequencies
- 14.4 Group Velocity
- 14.5 Summary
- 14.6 Exercises
- 14.7 References

Table of Contents cont.

- 15. Guided Waves in Anisotropic Media
- 15.1 Introduction
- 15.2 Phase Velocity Dispersion
- 15.3 Guided Wave Directional Dependency
- 15.4 Guided Wave Skew Angle
- 15.5 Guided Waves in Composites with Multiple

Layers - 15.6 Exercises
- 15.7 References

Table of Contents cont.

- 16. Guided Wave Phased Arrays in Piping
- 16.1 Introduction
- 16.2 Guided Wave Phased Array Focus Theory
- 16.3 Numerical Calculations
- 16.4 Finite Element Simulation of Guided Wave

Focusing - 16.5 Active Focusing Experiment
- 16.6 Guided Wave Synthetic Focus
- 16.7 Synthetic Focusing Experiment
- 16.8 Summary
- 16.9 Exercises
- 16.10 References

Table of Contents cont.

- 17 . Guided Waves in Viscoelastic Media
- 17.1 Introduction
- 17.2 Viscoelastic Models
- 17.2.1 Material Viscoelastic Models
- 17.2.2 Kelvin-Voight Model
- 17.2.3 Maxwell Model
- 17.2.4 Further Aspects of the Hysteretic and

Kelvin-Voight Models - 17.3 Measuring Viscoelastic Parameters
- 17.4 Viscoelastic Isotropic Plate
- 17.5 Viscoelastic Orthotropic Plate
- 17.5.1 Problem Formulation and Solution
- 17.5.2 Numerical Results
- 17.5.3 Summary
- 17.6 Lamb Waves in a Viscoelastic Layer
- 17.7 Viscoelastic composite Plate
- 17.8 Pipes with Viscoelastic Coatings
- 17.9 Exercises
- 17.10 References

Table of Contents cont.

- 18. Ultrasonic Vibrations
- 18.1 Introduction
- 18.2 Practical Insights into the Ultrasonic

Vibrations Problem - 18.3 Concluding Remarks
- 18.4 Exercises
- 18.5 References

Table of Contents cont.

- 19. Guided Wave Array Transducers
- 19.1 Introduction
- 19.2 Analytical Development
- 19.2.1 Linear Comb Array Solution
- 19.2.2 Annular Array Solution
- 19.3 Phased Transducer Arrays for Mode Selection
- 19.3.1 Phased Array Analytical Development
- 19.3.2 Phased Array Analysis
- 19.4 Concluding Remarks
- 19.5 Exercises
- 19.6 References

Table of Contents cont.

- 20. Introduction to Guided Wave Nonlinear Methods
- 20.1 Introduction
- 20.2 Bulk Waves in Weakly Nonlinear Elastic

Media - 20.3 Measurement of the Second Harmonic
- 20.4 Second Harmonic Generation Related to

Microstructure - 20.5 Weakly Nonlinear Wave Equation
- 20.6 Higher Harmonic Generation in Plates
- 20.6.1 Synchronism
- 20.6.2 Power Flux
- 20.6.3 Group Velocity Matching
- 20.6.4 Sample Laboratory Experiments
- 20.7 Applications of Higher Harmonic Generation

by Guided Waves - 20.8 Exercises
- 20.9 References

Table of Contents cont.

- 21. Guided Wave Imaging Methods
- 21.1 Introduction
- 21.2 Guided Wave through Transmission Dual Probe

Imaging - 21.3 A Defect Locus Map
- 21.4 Guided Wave Tomographic Imaging
- 21.5 Guided Wave Phased Array in Plates
- 21.6 Long-Range Ultrasonic Guided Wave Pipe

Inspection Images - 21.7 Exercises
- 21.8 References

Table of Contents cont.

Appendix A Ultrasonic Nondestructive Testing

Principles, Analysis, and Display Technology

- A.1 Physical Principles
- A.2 Wave Interference
- A.3 Computational Model for a Single Point Source
- A.4 Directivity Function for a Cylindrical

Element - A.5 Ultrasonic Field Presentations
- A.6 Near-Field Calculations
- A.7 Angle-of-Divergence Calculations
- A.8 Ultrasonic Beam Control
- A.9 A Note of Ultrasonic Field

Solution Techniques - A.10 Time and Frequency Domain Analysis
- A.11 Pulsed Ultrasonic Field Effects
- A.12 Introduction to Display Technology
- A.13 Amplitude Reduction of an Ultrasonic

Waveform - A.14 Resolution and Penetration Principles
- A.14.1 Axial Resolution
- A.14.2 Lateral Resolution
- A.15 Phase Arrays and Beam Focusing
- A.16 Exercises
- A.17 References

Table of Contents cont.

- Appendix B Basic Formulas and Concepts in the

Theory of Elasticity - B.1 Introduction
- B.2 Nomenclature
- B.3 Stress, Strain, and Constitutive Equations
- B.4 Elastic Constant Relationships
- B.5 Vector and Tensor Transformation
- B.6 Principal Stresses and Strains
- B.7 The Strain Displacement Equations
- B.8 Derivation of the Governing Wave Equation
- B.9 Anisotropic Elastic Constants
- B.10 References

Table of Contents cont.

- Appendix C Physically Based Signal Processing

Concepts for Guided Waves - C.1 General Concepts
- C.2 The Fast Fourier Transform (FFT)
- C.2.1 Example FFT Use Analytic Envelope
- C.2.2 Example FFT Use Feature Source for

Pattern Recognition - C.2.3 Discrete Fourier Transform Properties
- C.3 The Short Time Fourier Transform (STFFT)
- C.3.1 Example STFFT to Dispersion Curves
- C.4 The 2-D Fourier Transform (2DFFT)
- C.5 The Wavelet Transform (WT)
- C.6 Exercises
- C.7 References

Table of Contents cont.

- Appendix D Guided Wave Mode and Frequency

Selection Tips - D.1 Introduction
- D.2 Mode and Frequency Selection Considerations
- D.2.1 A Surface-Breaking Defect
- D.2.2 Mild Corrosion and Wall Thinning
- D.2.3 Transverse Crack Detection in the Head of

a Rail - D.2.4 Repair Patch Bonded to an Aluminum Layer
- D.2.5 Water-Loaded Structures
- D.2.6 Frequency and Other Tuning Possibilities
- D.2.7 Ice Detection with Ultrasonic Guided

Waves - D.2.8 Deicing
- D.2.9 Real Time Phased Array Focusing in Pipe
- D.2.10 Aircraft, Lap-Splice, Tear Strap, and

Skin to Core Delamination Inspection

Potential - D.2.11 Coating Delamination and Axial Crack

Detection - D.2.12 Multilayer structures
- D.3 Exercises
- D.4 References

Background

- Preface
- To start now with Chapter 1. Lets see a few

references first, of many listed in the book

after each chapter. - References
- Achenbach, J. D. (1976). Generalized continuum

theories for directionally reinforced solids,

Arch. Mech. 28(3) 25778. - Achenbach, J. D. (1984). Wave Propagation in

Elastic Solids. New York North-Holland. - Achenbach, J. D. (1992). Mathematical modeling

for quantitative ultrasonics, Nondestr. Test.

Eval. 8/9 36377. - Achenbach, J. D., and Epstein, H. I. (1967).

Dynamic interaction of a layer of half space, J.

Eng. Mech. Division 5 2742. - Achenbach, J. D., Gautesen, A. K., and McMaken,

H. (1982). Ray Methods for Waves in Elastic

Solids. Boston, MA Pitman. - Achenbach, J. D., and Keshava, S. P. (1967). Free

waves in a plate supported by a semi-infinite

continuum, J. Appl. Mech. 34 397404. - Auld, B. A. (1990). Acoustic Fields and Waves in

Solids. 2nd ed., vols. 1 and 2. Malabar, FL

Krieger. - Auld, B. A., and Kino, G. S. (1971). Normal mode

theory for acoustic waves and their application

to the interdigital transducer, IEEE Trans.

ED-18 898908.

Background cont.

- Auld, B. A., and Tau, M. (1978). Symmetrical Lamb

wave scattering at a symmetrical pair of thin

slots, in 1977 IEEE Ultrasonic Sympos. Proc. vol.

61. - Beranek, L. L. (1990). Acoustics. New York

Acoustical Society of America, American Institute

of Physics. - Davies, B. (1985). Integral Transforms and Their

Applications. 2nd ed. New York Springer-Verlag. - Eringen, A. C., and Suhubi, E. S. (1975). Linear

Theory (Elastodynamics, vol. 2). New York

Academic Press. - Ewing, W. M., Jardetsky, W. S., and Press, F.

(1957). Elastic Waves in Layered Media. New York

McGraw-Hill. - Federov, F. I. (1968). Theory of Elastic Waves in

Crystals. New York Plenum. - Graff, K. F. (1991). Wave Motion in Elastic

Solids. New York Dover. - Kino, C. S. (1987). Acoustic Waves Devices,

Imaging and Digital Signal Processing. Englewood

Cliffs, NJ Prentice-Hall. - Kinsler, L. E., Frey, A. R., Coppens, A. B., and

Sanders, J. V. (1982). Fundamentals of Acoustics.

New York Wiley. - Kolsky, H. (1963). Stress Waves in Solids. New

York Dover. - Love, A. E. H. (1926). Some Problems of

Geodynamics. Cambridge University Press. - Love, A. E. H. (1944a). Mathematical Theory of

Elasticity. 4th ed. New York Dover.

Background cont.

- Love, A. E. H. (1944b). A Treatise on the

Mathematical Theory of Elasticity. New York

Dover. - Miklowitz, J. (1978). The Theory of Elastic Waves

and Waveguides. New York North-Holland. - Mindlin, R. D. (1955). An Introduction to the

Mathematical Theory of Vibrations of Elastic

Plates. Fort Monmouth, NJ U.S. Army Signal Corps

Engineers Laboratories. - Musgrave, M. J. P. (1970). Crystal Acoustics. San

Francisco, CA Holden-Day. - Pollard, H. F. (1977). Sound Waves in Solids.

London Pion Ltd. - Rayleigh, J. W. S. (1945). The Theory of Sound.

New York Dover. - Redwood, M. (1960). Mechanical Waveguides. New

York Pergamon. - Rose, J. L. (1999). Ultrasonic Waves in Solid

Media. Cambridge University Press. - Rose, J. L. (2002). A baseline and vision of

ultrasonic guided wave inspection potential,

Journal of Pressure Vessel Technology 124

27382. - Stokes, G. G. (1876). Smiths prize examination,

Cambridge. Reprinted 1905 in Mathematics and

Physics Papers vol. 5, p. 362, Cambridge

University Press. - Viktorov, I. A. (1967). Rayleigh and Lamb Waves

Physical Theory Applications. New York Plenum.

Major Contributors

- Michael Avioli
- Cody Borigo
- Jason Bostron
- Huidong Gao
- Cliff Lissenden
- Yang Liu

- Vamshi Chillara
- Jing Mu
- Jason Van Velsor
- Fei Yan
- Li Zhang

Dedication Aleksander Pilarski

- Wave propagation studies are not limited to NDT

and SHM, of course. Many major areas of study in

elastic wave analysis are under way, including - (1) transient response problems, including

dynamic impact loading - (2) stress waves as a tool for studying

mechanical properties, such as the modulus of

elasticity and other anisotropic constants and

constitutive equations (the formulas relating

stress with strain and/or strain rate can be

computed from the values obtained in various,

specially designed, wave propagation

experiments) - (3) industrial and medical ultrasonics and

acoustic-emission nondestructive testing

analysis - (4) other creative applications, for example, in

gas entrapment determination in a pipeline, ice

detection, deicing of various structures, and

viscosity measurements of certain liquids and - (5) ultrasonic vibration studies that combine

traditional low-frequency vibration

analysis tools in structural analysis with

high-frequency ultrasonic analysis.

Figure 1-1 Comparison of bulk wave and guided

wave inspection methods.

Table 1.1 Ultrasonic Bulk vs. Guided Wave

Propagation Considerations

BULK GUIDED

Phase Velocities Constant Function of frequency

Group Velocities Same as phase velocities Generally not equal to phase velocity

Pulse Shape Non-dispersive Generally dispersive

- The principal advantages of using ultrasonic

guided waves analysis techniques can be

summarized as follows. - Inspection over long distances, as in the

length of a pipe, from a single probe position is

possible. Theres no need to scan the entire

object under consideration all of the data can

be acquired from the single probe position. - Often, ultrasonic guided wave analysis

techniques provide greater sensitivity, and thus

a better picture of the health of the material,

than data obtained in standard localized normal

beam ultrasonic inspection or other NDT

techniques, even when using lower frequency

ultrasonic guided wave inspection techniques. - Continued on next slide

- Continued from previous slide
- The ultrasonic guided wave analysis techniques

allow the inspection of hidden structures,

structures under water, coated structures,

structures running under soil, and structures

encapsulated in insulation and concrete. The

single probe position inspection using wave

structure change and wave propagation controlled

mode sensitivity over long distances makes these

techniques ideal. - Guided wave propagation and inspection are

cost-effective because the inspection is simple

and rapid. In the example described earlier,

there would be no need to remove insulation or

coating over the length of a pipe or device

except at the location of the transducer tool.

Table 1.2 Ultrasonic Wave Considerations for

Isotropic vs. Anisotropic Media

ISOTROPIC ANISOTROPIC

Wave Velocities Not function of launch direction Function of launch direction

Skew Angles No Yes

Table 1.3 - A Comparison of the Currently Used

Ultrasonic Bulk Wave Technique and the Proposed

Ultrasonic Guided Wave Procedure for Plate and

Pipe Inspection

Bulk Wave Guided Wave

Tedious and time consuming Fast

Point by point scan (accurate rectangular grid scan) Global in nature (approximate line scan)

Unreliable (can miss points) Reliable (volumetric coverage)

High level training required for inspection Minimal training

Fixed distance from reflector required Any reasonable distance from reflector acceptable

Reflector must be accessible and seen Reflector can be hidden

Table 1.4. Natural Waveguides

Plates (aircraft skin)

Rods (cylindrical, square, rail, etc.)

Hollow cylinder (pipes, tubing)

Multi-layer structures

An interface

Layer or multiple layers on a half-space

Table 1.5. The Difference between SHM and

Non-Destructive Testing (NDT)

- SHM
- On-line evaluation
- Condition based maintenance
- Determine fitness-for-service and remaining

useful time - Less cost and labor
- Baseline required
- Environmental data compensation methods are

required

- NDT
- Off-line evaluation
- Time base maintenance
- Find existing damage
- More cost and labor
- Baseline not available

Table 1.6. Successes Guided Waves in General

Increased computational efficiency developments and Understanding Basic Principles

Phased Array and Focusing developments in plates and pipes

Demonstration of optimal mode and frequency selections for penetration power, fluid loading influences, and other defect detection sensitivity requirements

Table 1.7 Successes Composite Materials

Understanding guided wave behavior in anisotropic media ( Slowness profiles and Skew angle influence)

Development of ultrasonic guided wave tomographic imaging methods

Comb sensor designs for optimal mode and frequency selection (linear comb and annular arrays)

Table 1.8 Successes Aircraft Applications

Demonstration of feasibility studies in composites and lap splice, tear strap, skin to core delamination, corrosion detection and other applications.

Table 1.9 Successes Pipe Inspection

Understanding and utilization of both axisymmetric and non-axisymmetric modes

Achieving excellent penetration power with special sensors, focusing, and mode and frequency choices

Handling fluid loading with Torsional Modes

Defect sizing accomplishments to less than 5 cross sectional area

Reduced false alarm calls in inspection due to focusing for confirmation

Circumferential location and length of defect estimations with focusing

Testing of Pipe under insulation, coatings, and/or soil

Table 1.10 Practical Challenges Guided Waves in

general

Modeling accuracy is critically dependent on accurate input parameters often difficult to obtain (especially for anisotropic and viscoelastic properties, interface conditions, and defect characteristics.)

Signal interpretations often difficult (due to multimode propagation and mode conversion, along with special test structure geometric features)

Sensor robustness to environmental situations like temperature, humidity to high stress, mechanical vibrations, shock and radiation

Adhesive bonding challenges for mounting sensors and sustainability in an SHM environment

Merger of guided wave developments with energy harvesting and wireless technology

Penetration power requirements

Table 1.11 Practical Challenges Composite

Materials

Dealing with complex anisotropy and wave velocity and skew angle as a function of direction

Viscoelastic influences

Penetration power due to anisotropy, viscoelasticity, and inhomogeneity

Differentiating critical composite damage such as delamination defects from structural variability during fabrication (including minor fiber misalignments, ply-drops, inaccurate fiber volume fraction, and so on)

Guided wave inspection of composites with unknown material properties.

Table 1.12 Practical Challenges Aircraft

Applications

Robustness of guided wave sensors under in-flight conditions

Influences of aircraft paint and embedded metallic mesh in composite airframes for lightning protection

Table 1.13 Challenges Pipe Inspection

Tees, elbows, bends, and number of elbows and inspection beyond elbows

Quantification in defect location, characterization, sizing, especially depth determination

Inspection reliability and false alarms (due to multimode propagation, mode conversions, and so many pipe features like welds, branches, etc.)

Reducers, expanders, unknown layout drawings, cased pipes and sleeves