This book presents basic structural deformation and stress analysis as applied to optical systems. It provides the tools for first-order analyses required in the design concept phase before handling the intricate details of a full-up design. While finite element analysis is paramount to a successful design, the purpose of this text is not to use finite element analysis to validate the hand analysis, but rather to use hand analysis to validate the finite element models. The hand analysis forces a discipline that is paramount in the understanding of structural behavior. Presuming that the reader has a working knowledge in the strength of materials, the text applies engineering principles to opto-structural analysis.

### Table of Contents

*Preface**Acknowledgments**A Note on Units***1 Stress and Strain**- 1.1 Introduction
- 1.2 Hooke's Law
- 1.3 Beyond Tension, Compression, and Shear
- 1.3.1 Bending stress
- 1.3.2 Bending deflection
- 1.3.3 Shear stress due to bending
- 1.3.4 Shear deflection due to bending (detrusion)
- 1.3.5 Torsion
- 1.3.6 Hooke's law summary
- 1.4 Combining Stresses
- 1.4.1 Bending stress
- 1.5 Examples for Consideration
- 1.6 Thermal Strain and Stress
- 1.6.1 Thermal hoop stress
- 1.6.2 Ring in ring in ring
- 1.6.3 Nonuniform cross-section
- 1.7 Buckling
- References
**2 Material Properties**- 2.1 Properties and Definitions
- 2.2 Low-Thermal-Expansion Materials
- 2.2.1 Fused silica
- 2.2.2 ULE® fused silica
- 2.2.3 ZERODUR®
- 2.2.4 Silicon
- 2.2.5 Silicon carbide
- 2.2.6 Graphite composites
- 2.2.7 Invar®
- 2.2.8 Iron-nickel varieties
- 2.2.9 The iron-nickel family
- 2.2.10 Governing specifications
- 2.2.11 Invar summary
- 2.3 Not-So-Low-Thermal-Expansion Materials
- 2.3.1 Aluminum,
- 2.3.2 Beryllium
- 2.3.3 Aluminum-beryllium
- 2.3.4 Optical metering
- 2.4 Very High-Thermal-Expansion Materials
- 2.4.1 Plastics
- 2.4.2 Adhesives
- 2.5 Strength
- 2.5.1 Failure to load
- 2.5.2 Yield
- 2.5.3 Micro-yield
- 2.5.4 Brittle materials
- 2.5.5 Safety factor
- 2.5.6 Summary
- References
**3 Kinematic Mounts**- 3.1 Kinematics
- 3.2 Quasi-static Kinematic Mount
- 3.3 Flexure Analysis
- 3.3.1 Rotational compliance about a radial line
- 3.3.2 Analysis: constrained degrees of freedom
- 3.3.3 Analysis: compliant degrees of freedom
- 3.4 Bipod
- 3.4.1 Analysis: constrained degree of freedom
- 3.4.2 Analysis: compliant degrees of freedom
- 3.5 Timmy Curves
- 3.5.1 Examples
- 3.5.2 Other effects
- 3.6 A Better Bipod
- 3.6.1 Analysis: constrained degrees of freedom
- 3.6.2 Analysis: compliant degrees of freedom
- 3.6.3 Example for reconsideration
- 3.7 An Alternative Bipod
- 3.8 Stroke Algorithm
- References
**4 Solid Optics: Performance Analysis**- 4.1 Wavefront Error and Performance Prediction
- 4.2 Mount-Induced Error
- 4.2.1 Tangential moment
- 4.2.2 Radial load
- 4.2.3 Example for consideration
- 4.2.4 Radial and axial moments
- 4.3 Gravity Error
- 4.3.1 Optical axis vertical
- 4.3.2 Optical axis horizontal
- 4.3.3 Zero-gravity test
- 4.3.4 Other angles
- 4.3.5 Brain teaser
- 4.4 Temperature Soak
- 4.5 Thermal Gradient
- 4.5.1 Examples for consideration
- 4.5.2 Nonlinear gradients
- 4.5.3 Examples for consideration
- 4.5.4 Other gradients
- 4.6 Coating and Cladding
- 4.6.1 Examples
- 4.7 Rule of Mixtures
- 4.7.1 Two layers
- 4.7.2 Multiple layers
- 4.8 Trimetallic Strip
- 4.8.1 Example
- 4.9 Random Variations in the Coefficient of Thermal Expansion
- 4.9.1 Example
- References
**5 Lightweight Optics: Optimization**- 5.1 Lightweight Optics
- 5.2 Core Shape
- 5.2.1 Core geometry
- 5.2.2 Example
- 5.3 Core Stiffness
- 5.4 Partially Closed-Back Optics
- 5.5 Polish
- 5.5.1 Example
- 5.5.2 Advanced polish
- 5.6 Weight Optimization
- 5.6.1 Example
- 5.7 Stiffness Criteria
- 5.7.1 Examples
- 5.8 Stiffness Optimization
- 5.9 The Great Debate
- 5.9.1 Closed-back geometry
- 5.9.2 Open-back geometry
- 5.9.3 Open- and closed-back design comparisons
- 5.9.4 Shear deflection
- 5.9.5 Anisotropy
- 5.9.6 Analytical comparison
- 5.9.7 And the winner is . . .
- References
**6 Lightweight Optics: Performance Error**- 6.1 Mount-Induced Error
- 6.1.1 Tangential moment
- 6.1.2 Radial and axial moments
- 6.2 Gravity
- 6.2.1 Optical axis vertical
- 6.2.2 Optical axis horizontal
- 6.3 Gradients
- 6.3.1 Nonlinear temperature gradients
- 6.3.2 Example
- 6.4 Coating and Cladding
- 6.4.1 Quilt error
- 6.5 Random Variations in the Coefficient of Thermal Expansion
- 6.6 All Shapes and Sizes
- 6.6.1 A case study
- References
**7 Large Optics**- 7.1 Multipoint Mounts
- 7.1.1 Example for consideration
- 7.2 Zonal Mount
- 7.3 Hindle Mount
- 7.4 Active Mount
- 7.4.1 Active-mount correctability illustration
- 7.4.2 An active-mount mechanism
- 7.5 Large-Aspect-Ratio Optics
- 7.5.1 Funny things happen at infinity
- 7.5.2 How large is large?
- 7.5.3 Cladding
- 7.5.4 Coating
- 7.5.5 Humidity
- 7.5.6 Thermal soak CTE variation
- 7.5.7 Thermal gradient
- 7.5.8 Metrology
- 7.5.9 Gravity
- 7.5.10 Edge machining
- 7.5.11 Delayed elasticity
- 7.6 Performance Comparisons
- 7.7 How Low Can You Go?
- 7.8 Extremely Large-Aspect-Ratio Optics
- 7.9 Summary
- References
**8 Figures of Merit**- 8.1 Mechanical Figures of Merit
- 8.2 Thermal Figure of Merit
- 8.3 Combined Figures of Merit
- 8.4 True Mechanical Figures of Merit
- 8.4.1 Weight and performance figures of merit
- 8.5 Strength-to-Weight Ratio
- 8.5.1 Gravitational acceleration: bending
- 8.5.2 External bending load and gravity acceleration
- 8.6 Graphical Summary
- 8.7 Lightweight Optics
- 8.8 Examples
- References
**9 Adhesives**- 9.1 Mechanical Properties
- 9.1.1 Elastic modulus
- 9.1.2 Static strength
- 9.1.3 Peel strength
- 9.2 Load Stress Distribution
- 9.3 Glass–Liquid Transition
- 9.3.1 Example for consideration
- 9.4 Temperature Creep
- 9.5 Lap shear strength
- 9.5.1 Surface preparation
- 9.6 Thermal Stress
- 9.6.1 Thermal stress at boundaries
- 9.7 Modeling Techniques
- 9.7.1 Element size
- 9.7.2 Thermal stress
- 9.8 Fillets
- 9.9 Soft Elastomers
- References
**10 Simple Dynamics**- 10.1 Basics
- 10.2 A Useful Relationship
- 10.2.1 Rotational frequency
- 10.2.2 Example
- 10.3 Random Vibration
- 10.3.1 Example
- 10.3.2 Decibels
- 10.4 Force Limits
- 10.4.1 Response limiting
- 10.5 Shipping Vibration
- 10.5.1 Drop shock
- 10.6 Acceleration Shock
- 10.6.1 Example
- 10.6.2 Variable acceleration
- 10.6.3 Lift equipment
- 10.6.4 Pyrotechnic shock
- References
**11 Fatigue**- 11.1 Cyclic Fatigue
- 11.1.1 High-cycle fatigue
- 11.2 S-N Method
- 11.2.1 Example for consideration
- 11.3 Nonzero Mean Stress
- 11.3.1 Example
- 11.3.2
*R*ratio - 11.4 Fracture Mechanics Method
- 11.4.1 Stress intensity
- 11.4.2 I love Paris
- 11.4.3 Case study
- 11.5 Random Vibration Fatigue
- 11.5.1 Miner's rule: discrete
- 11.5.2 Miner's rule: continuous
- 11.5.3 Multiple degrees of freedom
- References
**12 Brittle Materials**- 12.1 Theoretical Strength
- 12.2 Failure Modes
- 12.2.1 Mode I failure description
- 12.2.2 Residual stress
- 12.3 Strength Theory
- 12.3.1 General strength equation: residual stress free
- 12.3.2 Finite bodies and free-surface correction
- 12.3.3 General point flaws
- 12.3.4 The basic fracture mechanics equation
- 12.3.5 Example for consideration
- 12.4 Strength with Residual Stress
- 12.4.1 Combined residual stress and applied stress
- 12.4.2 Crack stability
- 12.4.3 Strength with residual stress and applied stress
- 12.4.4 Example for consideration
- 12.5 Stress Corrosion
- 12.5.1 Definitions
- 12.5.2 Chemically active environment
- 12.5.3 Reaction rates
- 12.5.4 I love Paris
- 12.5.5 Crack growth regions
- 12.5.6 Region I relation
- 12.5.7 Example for consideration
- 12.6 Stress Corrosion Free of Residual Stress
- 12.6.1 Examples for consideration
- 12.7 Stress Corrosion with Residual Stress
- 12.7.1 A complex integration
- 12.7.2 Computation of constants and resulting time to failure
- 12.7.3 Examples for consideration
- 12.7.4 Obtaining constants and failure time
- 12.8 Dynamic Fatigue
- 12.8.1 Example for consideration
- 12.9 An Approximation Technique
- 12.10 Overload Proof Test
- 12.10.1 Application to ceramics
- 12.10.2 Examples for consideration
- References
**13 Performance Analysis of Optical Structures**- 13.1 Supporting Optics
- 13.2 Metering Despace
- 13.2.1 Example for consideration
- 13.3 Decentration and Tip
- 13.3.1 Example for consideration
- 13.3.2 Gravity and frequency
- 13.4 Structure Forms
- 13.5 Metering Truss Design
- 13.5.1 Serrurier truss
- 13.5.2 Thermal expansion
- 13.5.3 Athermalized truss: a design before its time
- 13.5.4 Composite metering structure
- 13.6 Case Study: Teal Ruby Telescope
- 13.7 Support Structure
- References
**14 Nuts and Bolts**- 14.1 Terminology
- 14.2 Bolt Material
- 14.3 Bolt Stress
- 14.3.1 Shear
- 14.3.2 Thread shear
- 14.4 Stress Examples
- 14.5 Bolt Load
- 14.5.1 Preload
- 14.5.2 Externally applied load
- 14.5.3 External load vibration: bolt fatigue
- 14.5.4 Example for consideration
- 14.6 Thermal Load
- 14.6.1 Examples
- 14.7 Washers
- 14.7.1 Flat washers
- 14.7.2 Lock washers
- 14.7.3 Lock nuts
- 14.7.4 Locking and staking
- 14.7.5 Spring washers
- 14.8 Friction Slip and Pins
- 14.8.1 Friction
- 14.8.2 Pins
- 14.8.3 Shear tearout
- 14.8.4 Example for consideration
- 14.9 Combined Bolt Loads
- 14.9.1 The bolt circle
- References
**15 Linear Analysis of Nonlinear Properties**- 15.1 Linear Theory
- 15.2 Nonlinear Systems: Secant and Tangent Properties
- 15.2.1 Thermal expansion coefficient
- 15.2.2 Elastic modulus
- 15.3 Nonlinear Modulus
- 15.4 Nonlinear Thermal Stress
- 15.5 Special Theory
- 15.5.1 Constant CTE
- 15.5.2 Constant modulus
- 15.6 General Theory
- 15.6.1 Example for consideration
- 15.7 Using Secants
- 15.8 Sample Problems
**16 Miscellaneous Analysis**- 16.1 Venting
- 16.1.1 Contaminants
- 16.2 Stress Birefringencet
- 16.2.1 Coating-induced birefringence
- 16.2.2 Residual stress
- 16.3 Bonded Tubes and Grooves
- 16.3.1 Bending moment
- 16.3.2 Axial load
- 16.3.3 Torsion
- 16.3.4 Shear
- 16.3.5 Tube over boss
- 16.3.6 Square boss
- 16.4 Bonded Flexures
- 16.4.1 Example for consideration
- 16.5 Contact Stress
- 16.5.1 Ball-on-flat formulation
- 16.5.2 Ball-in-cone formulation
- 16.5.3 Ball-in-cone analysis
- 16.5.4 Kinematic coupling
- 16.5.5 Allowable load: Hertzian stress
- 16.6 Friction
- 16.6.1 Surface roughness
- 16.7 Large Displacements
- 16.8 Windows
- 16.8.1 Bending
- 16.8.2 Lateral thermal gradient
- 16.9 Dimensional Instability
- 16.9.1 Glass transition temperature
- 16.9.2 Hysteresis
- 16.9.3 External stress relation
- 16.9.4 Creep
- 16.9.5 Glass and ceramics
- 16.9.6 Invar 36
- 16.9.7 Internal (residual) stress
- 16.9.8 Metal optics
- References
*Epilogue**Index*

## Preface

Texts on structural and mechanical analysis are numerous, and indeed, this
entire text is based on the pioneering works of others in the field. This book,
therefore, draws on those texts and presumes a working knowledge of the
strength of materials [see J. W. Pepi, *Strength Properties of Glass and
Ceramics*, SPIE Press (2014)]. With that foundation, we apply those
engineering principles to opto-structural analysis. In the precision world of
optics, we are often concerned with displacements and deformations of very
small values, from fractions of a wavelength of light to the micron and
nanometer (millionths of an inch) order. Furthermore, optical systems
designed for flight are often required to be of very light weight. While the
analytical techniques in any case are the same as on the macro level, careful
analysis is required when moving the decimal point so far to the left.

*don't*move once they are deformed, mechanical analysis as applying to things that

*move*(such as mechanisms), and dynamic analysis as applying to things that

*move slightly*, the title selection becomes more clear (although these latter topics are discussed in the book).

This book is written with the intent to understand basic structural deformation and stress analysis as applied to optical systems. It provides the tools for first-order analyses required in the design concept phase before entering into the intricate details of a full-up design. Ever-increasing computer technology has allowed former tedious and unwieldy problems to be solved in a fraction of the time by using finite element analysis. Unfortunately, reliance on such fast methods without hand analysis backup can lead to unsuspected errors. Thus, first-order calculations are an excellent way to complement the current state of the industry that relies more on computational design techniques. These calculations accelerate the design process by allowing an understanding of the critical governing parameters and allowing accelerated design trades and sensitivity studies to be performed that decrease schedule and cost. The insights gained from these techniques can then be used to guide the development of appropriate finite element models, including model fidelity, and details focusing on the critical and most sensitive design parameters. These models, in turn, are more efficient and provide the optostructural engineer a comprehensive and insightful design approach. This approach can then inform the roadmap for risk reduction and environmental testing.

While finite element analysis is paramount to a successful design, the purpose of this text is not to use finite element analysis to validate the hand analysis but rather to use hand analysis to validate the finite element models. The hand analysis forces a discipline that aids tremendously in the understanding of structural behavior. It is the intent, then, not to forget such techniques."Forsan et haec olim meminisse iuvabit."*

*From Virgil's The Aeneid [translation: "Perhaps, someday, we will look back fondly on these things."]

## Acknowledgments

Nothing can be learned or known without the pioneering efforts of others. It is with deep gratitude that I acknowledge the work of Stephen Timoshenko, the father of engineering mechanics, whose technical brilliance and straightforward communication skills have set the groundwork for this book.I would also like to acknowledge the many teachers, professors, supervisors, and peers who have assisted me through the years, without whom this work would not be possible. A special note of thanks is given to Paul Yoder, Jr., who, sadly, is deceased, for his encouragement to produce this text and to Dan Vukobratovich for his insight and expertise.

I am very much indebted to Stefanos Axios, mechanical engineer, for his preparation, editing, and checking of the multitude of equations herein presented, and for his diligence, suggestions, and critique of the manuscript. I would particularly like to acknowledge and dedicate this book to Francis G. Bovenzi and Joseph E. Minkle, supervisors at Itek Optical Systems, Lexington, Massachusetts, who taught me much of what I know.Finally, I thank my wife, Sandy, for both her patience and encouragement in the preparation of this text.

**John W. Pepi**
October 2018

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