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Spie Press Book

Opto-structural Analysis
Author(s): John W. Pepi
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Book Description

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.

Book Details

Date Published: 10 December 2018
Pages: 480
ISBN: 9781510619333
Volume: PM288

Table of Contents
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Table of Contents

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

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

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

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

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 . . .

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

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

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

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

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

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

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

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

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

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



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.

In preparing to write this book, some thought went into the title. Before selecting the term "opto-structural analysis," an alternative term "optomechanical analysis" was considered for the title. However, several excellent texts under that latter title are available. It is certainly not the intent to replace those worthy sources but rather to supplement them. To this end, the title contains the term "opto-structural," perhaps because the author is a structural engineer, but more so, to point out the "static" nature of the topic. If structural analysis is defined as applying to things that 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."]


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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