Aircraft Structures for Engineers

(1)

Aircraft Structures for Engineers

Vijay K. Goyal, Ph.D.

Associate Professor, Mechanical Engineering Department, University of Puerto Rico at Mayag¨uez,

Mayag¨uez, Puerto Rico

Vinay K. Goyal, Ph.D.

Sr. Member of the Technical Staff, Structural Mechanics Subdivision, The Aerospace Corporation,

Los Angeles, California

M.S. / M.Eng. in Mechanical Engineering / Minor in Aerospace Engineering (for B.S.)

This material is only for students enrolled at UPRM. All others must request permision from the author. ([email protected]

)

(2)

To the Almighty God: Father, Son Yeshua, and the Holy Spirit; Vijay: my wife Maricelis and my family Vinay: my wife Stacey and my family

“I can do all things in Yeshua who strengthens me...” – BIBLE: Phillipians 4:13

(3)

Aerospace Structures is one of the most challenging courses to teach. It enclosed many advanced topics while introducing some fundamental thin-walled structural analysis. This book is written for students with a background in mechanical engineering, although the concepts are presented in a funda-mental approach allowing students from all backgrounds to benefit from the material in this book. Intended Audience

This book is intended to provide a foundation of the finite element and optimization techniques. The target audience are senior level undergraduate and first year graduate students who have had little, or no, exposure to thin-walled structural analysis. Practicing engineers will also benefit from the integration approach to obtain very impressive and useful results. Thus, we can assure that this book will fill up a void in the personal library of many engineers who are trying to, or planning, to design and analyze thin-walled structures. A background in solid mechanics, calculus, and basic programming knowledge is required.

Motivation

When writing this textbook, we have kept the reader in mind at all times. After years of using this manuscript, engineering graduates (from the University of Puerto Rico at Mayag¨uez) have found the manuscript very useful in their respective jobs. In teaching and applying these subjects for years, we have come to the conclusion that students and engineers too often take a ”black-box” approach. The book also tries to bind traditional theoretical approaches with some modern nu-merical techniques. The original is proven to be an effective reference in the aerospace industry, such as Boeing and InfoTech Aerospace Services. The format of this book is student-friendly since each chapter begins with instructional objectives and ends with a chapter summary highlighting the most important aspects of the chapter with an outline of ongoing research within the top-ics presented in the chapter. The authors assume that the students have little experience with programming languages and numerical methods; thus, this is a reader-friendly book that enables the reader to self-learn the topics. It includes a variety of examples, specifically worked with a pedagogical approach, using a step-by-step procedure which is easy to apply to a wide range of engineering problems. At the end of each chapter one can find a variety of problems that are been carefully worked-out in an accompanying solution manual to the textbook, available online to the instructors. Importance was given to emphasis on application to keep the students interested in the subject. After the reader has completed this book, he/she will be able to:

1. Understand how and why the aircraft are designed the way they are.

2. Learn and apply the fundamentals of the linear elasticity to the analysis of thin-walled struc-tures.

3. Use numerical techniques to approximate analytical solutions.

(4)

Mathematical Level

Readers need a strong background in linear algebra, calculus, differential equations, and program-ming. For those who have not been exposed to linear algebra, we have included an appendix that will enable the reader to self-study, or review, this topic. The authors assume that the readers have little experience with programming languages and numerical methods.

Chapter Organization and Topical Coverage

The format of this book is student-friendly since each chapter begins with instructional objectives and ends with a chapter summary highlighting the most important aspects of the chapter with an outline of ongoing research within the topics presented in the chapter. It includes a variety of examples, specifically worked with a pedagogical approach, using a step-by-step procedure which is easy to apply to a wide range of engineering problems. At the end of each chapter one can find a variety of problems that have been carefully worked-out in an accompanying solution manual to the textbook, available online to the instructors. Emphasis was placed on applications to keep the reader interested in the subject.

The contents of this book are intended for a two semester term course. All examples have been solved using Mathematica and MATLABr _{(which are available to students and instructors}

through the book website).

In short, this unique book will help the reader, whether a student or a practicing engineer, to independently learn the topics through carefully worked out examples and apply them to real aircraft engineering design problems. Any comments and suggestions can be sent to [email protected].

Vijay K. Goyal July 23, 2008

(5)

1. Instructor

1a. Dr. Vijay K. Goyal, Associate Professor of the Mechanical Engineering Department 1b. Office: L-207

1c. Office Hours: MW: 2:30p - 3:30p; Tu Th: 2:00p - 3:00p or by appointment

1d. Office Phone: (787) 832-4040 ext. 2111/3659 (Please do not call at home nor at my cell phone) 1e. E-mail: [email protected], [email protected]

2. General Information

2a. Course Number: INME 4717–5717

2b. Course Title: Aircraft Structural Analysis and Design Courses 2c. Credit-Hours: Three of lecture and lab included

2d. Classroom: L-236A 3. Course Description

Aircraft Structural Analysis and Design (INME4717): Application of solid mechanics to analyze aerospace structures. Study of aircraft components and their design philosophy. Applica-tion of elasticity to describe the stress, strain, and displacement fields of one- and two-dimensional problems in aerospace structures. Exact and approximate solutions of two-dimensional structural problems. Analysis of bending, shear and torsional theories for arbitrary, multimaterial, and mul-ticell wing cross-sections. Analysis of thin-walled single and mulmul-ticell stiffened shell beams using analytical and numerical solutions.

Advanced Aircraft Structural Design (INME5717): Application of work and energy prin-ciples, and numerical methods, to the design of flight vehicles. Study of deflection and load analysis using the Principle of Virtual Work, Principle of Complementary Virtual Work, analytical weak form solutions, and the finite element formulation. Wing design considering: fatigue, aeroelasticity, divergence, environmental loads, aerospace materials, dynamic stability of thin-walled compression members, and structural dynamics.

4. Pre/Co-requisites

Aircraft Structural Analysis and Design (INME4717):

4a. Pre-requisites: Design of Machine Elements I (INME 4011) or DIR AUTHORIZATION Advanced Aircraft Structural Design (INME5717):

4a. Pre-requisites: Aircraft Structural Analysis and Design (INME 4717) or DIR AUTHORIZA-TION

(6)

5. Textbook, Supplies and Other Resources

5a. Class notes are posted on the class website. The official course textbook is the course website: http://www.me.uprm.edu/vgoyal/inme4717.html

http://www.me.uprm.edu/vgoyal/inme5717.html 5b. Other useful references:

(a) Allen, D. H., Introduction to Aerospace Structural Analysis, 1985, John Wiley and Sons, New York, NY*.

(b) Curtis, Howard D., Fundamentals Of Aircraft Structural Analysis, First Edition, 1997, Mc-Graw Hill, New York, NY*

(c) Johnson, E. R., Thin-Walled Structures, 2006, Textbook at Virginia Polytechnic Institute and State University, Blacksburg, VA.

(d) Keane, Andy and Nair, Prasanth, Computational Approaches for Aerospace Design: The Pursuit of Excellence, August 2005, John Wiley and Sons.

(e) Newman, D., Interactive Aerospace Engineering And Design With CD-ROM, First Edi-tion, Mass Institute Of Tech, 2004, Mcgraw-Hill.

(f) Sun, C. T., Mechanics of Aircraft Structures, Second Edition 2006, John Wiley and Sons (g) Thomas, G. B., Finney R. L., Weir, M. D., and Giordano F. R., Thomas Calculus, Early

Transcendentals Update, 2003, Tenth Edition, Addison-Wesley, Massachusetts. Entire book

6. Purpose

Aircraft Structural Analysis and Design (INME4717): After completing this course stu-dents should be able to: (i) identify and understand the function of typical aircraft components, and discuss the behavior of monocoque and semi-monocoque structures; (ii) formulate multi-directional internal loads; (iii) formulate and analyze the state of point and state of stress; (iv) identify and evaluate various stress-strain formulations; (v) apply Hooke’s law including thermal effects; (vi) apply Euler-Bernoulli beam theory and Timoshenko beam theory; (vii) apply Airy Stress Func-tion; (viii) apply the classical torsional theory for prismatic beams; (ix) analyze bending, shear, and torsion of arbitrary, multicell and multimaterial cross-sectional wings.

Advanced Aircraft Structural Design (INME5717): After completing this course students should be able to: (i) analyze load and deflections of statically determinate and indeterminate structures using the Principle of Virtual Work and Principle of Complementary Virtual Work; (ii) analyze wings using the weak form and the finite element method; (iii) analyze and design wings based on fatigue, aeroelasticity, divergence, structural dynamics, and dynamic stability; (iv) learn and integrate environmental loads into aerospace design.

7. Course Goals

The course will be divided into four modules. Each module has the purpose to help the student understand and grasp the basic concept in computer aided design for mechanical engineering problems. See website.

7a. ( %) Chapter 1. Learning About Aircraft Structures 7b. ( %)

(7)

7d. ( %)

In addition to the above topics, all students will demonstrate the ability analyze a recent journal paper provided by the instructor: the context of the report (introduction), describe clearly and precisely the procedures used (methodology), report verbally and visually the findings (results), interpret the findings (analysis of results), justify the solutions persuasively (conclusions), and pro-vide final comments. The students will demonstrate the ability to make effective oral presentations and written reports using appropriate computer tools.

8. Requirements

8a. Requirements: In order to succeed in the course students are expected to: • should attend all class sessions and be punctual

• on a daily basis check the class website • use a non-programmable calculator • do all homework

• practice all suggested problems • take all exams

• submit all work in English

• be ready to ask any questions at the beginning of every class session • and obtain a minimum of 69.50% in the course

8b. Grading Distribution: Total course points are 100% and are distributed as follows:

Test I 25%

Test II 25%

Test III 25%

Test IV 25%

Students should take advantage of bonus homework and projects to improve their grade because there will be no “grade curving” at the end of the semester. Your grade will be determined by the following fixed grade scale:

A 89.500 − 100+

B 78.500 − 89.499

C 69.500 − 78.499

D 59.500 − 69.499

F 0 − 59.499

Your final grade will be scaled based on the attendance. For an example, if you miss 3 classes and your final grade is 100% then your official final grade will be 100*(42/45)=93%.

8c. Students failing to provide a successful, high-standard, computer projects may not pass the course, as they are entitled to a grade of IF or D, regardless of their progress in the mid-term examinations, homework, small projects, among other evaluation criteria. By successful we mean obtaining a percentage higher than 80% in overall projects. Moreover, a successful projects do not entitle the student to pass the course either (see 8b).

(8)

8d. Homework and Tests: Only your own handwritten solutions, written legibly on one side of an 8.500_{× 11}00_{sheet of paper will be accepted for grading. In the case of computer assignment, a}

computer print out is acceptable whenever a copy of the code is included and well documented by hand. Students are encouraged to work together on the homework, but submissions must be the student’s own work. NO LATE HOMEWORK WILL BE ACCEPTED.

9. Laboratory/Field Work (If applicable)

9a. Cell phones/pagers: All students MUST turn off their cell phones and pagers at the beginning of each class session. By not doing so it is considered disrespectful and students will be asked to leave the class. Students who need to have their cell phones or pagers on at all time must inform the instructor at the beginning of the academic semester.

9b. Smoking: Smoking is not permitted in any area other than those areas designated for smoking. 9c. Electronic Devices: Radios, tape recorders, and other audio or video equipment are not permitted in the lab or classroom at any time. Students must consult with the professor at the beginning of the academic semester.

9d. Laptop Computers, Notebooks, PC-Tablets: Students can bring their personal computers to classroom. However this must not interfere with other student’s work nor with the class session. Students with their personal computers are responsible for any problems with software versions or differences with the one available in the classroom.

10. Department/Campus Policies

10a. Class attendance: Class attendance is compulsory. The University of Puerto Rico at Mayag¨uez reserves the right to deal at any time with individual cases of non attendance. Professors are expected to record the absences of their students. Absences affect the final grade, and may even result in total loss of credits. Arranging to make up work missed because of legitimate class absence is the responsibility of the student. (Bulletin of Information Undergraduate Studies)

Students with three unexcused absences or more may be subject to a one or two final grade letter drop.

10b. Absence from examinations: Students are required to attend all examinations. If a student is absent from an examination for a justifiable and acceptable reason to the professor, he or she will be given a special examination. Otherwise, he or she will receive a grade of zero of “F” in the examination missed. (Bulletin of Information Undergraduate Studies)

In short, any student missing a test without prior notice or unexcused absence will be required to drop the course. There will be no reposition exam. At professor’s judgment, those students with a genuine excuse will be given an oral 15–20 minutes oral comprehensive final exam and it will substitute the missed examination(s).

Under no circumstances should the students schedule interviews during previously set dates for examinations.

10c. Final examinations: Final written examinations must be given in all courses unless, in the judgment of the Dean, the nature of the subject makes it impracticable. Final examina-tions scheduled by arrangements must be given during the examination period prescribed in

(9)

the Academic Calendar, including Saturdays. (see Bulletin of Information Undergraduate Studies).

Because of the nature of this course, the comprehensive final exam is substituted with the course project. During their oral presentation, students should be ready to answer any question(s) from the course.

10d. Partial withdrawals: A student may withdraw from individual courses at any time during the term, but before the deadline established in the University Academic Calendar. (see Bulletin of Information Undergraduate Studies).

10e. Complete withdrawals: A student may completely withdraw from the University of Puerto Rico at Mayag¨uez at any time up to the last day of classes. (see Bulletin of Information Undergraduate Studies).

10f. Disabilities: All the reasonable accommodations according to the Americans with Disability Act (ADA) Law will be coordinated with the Dean of Students and in accordance with the particular needs of the student. Those students with special needs must identify themselves at the beginning of the academic semester (with the professor) so that he/she can make the necessary arrangements according to the Office of Affairs for the Handicap. (Certification #44)

10g. Ethics: Any academic fraud is subject to the disciplinary sanctions described in article 14 and 16 of the revised General Student Bylaws of the University of Puerto Rico contained in Certification 018-1997-98 of the Board of Trustees. The professor will follow the norms established in articles 1-5 of the Bylaws. The honor code will be strictly enforced in this course. Students are encouraged to review the honor system policy which has been placed on the class website.

All assignments submitted shall be considered graded work unless otherwise noted. Thus all aspects of the course work are covered by the honor system. Any suspected violations of the honor code will be promptly reported to the honor system. Honesty in your academic work will develop into professional integrity. The faculty and students of UPRM will not tolerate any form of academic dishonesty. MUST BE TAKEN SERIOUSLY. Any violation may result in an automatic “F” in the course and such behavior will be reported to the Dean’s office of the College of Engineering.

(10)

11. General Topics

11a. Exam and Presentation Dates: (These dates may be subject to change) Test 1:

Module

Review session: Class Time

Exam Date: Posted on class website Test 2:

Module

Exam Date: Posted on class website Test 3:

Module

Exam Date: Posted on class website Final Project:

Presentations (Attendance compulsory) Date and Location to be posted on the website 11b. Course Outline:

(11)

There are many people who have made this work possible. First and foremost, I am mostly thankful to Yeshua, for giving me the opportunity to live in this time. All my success I give to him for He has been my strength and inspiration at all times.

Secondly, I express my special appreciation to my wife Maricelis, my son Jeremiah, and my daughter Naarah for their support and inspiration behind this effort. I could not have completed this task without their prayers, love, understanding, encouragement, and support.

Thirdly, I would like to thank all the graduate students who collaborated to complete this book: Juan Rein´es and Angel Quintero. In addition, many thanks to the invaluable inputs from the undergraduate students who used the manuscript form of this book during the 2004–2008 period at the University of Puerto Rico at Mayag¨uez.

Lastly, I cannot leave behind all the people who have given their suggestions to this work, such as Dr. Paul Sundaram, whom I consider my mentor. Special thanks to all the friends who encouraged and helped me achieve this goal.

God bless and thank you all,

Vijay K. Goyal

(12)

List of Figures xxi

List of Tables xxvii

Chapter 1. Learning about Aircraft Structures 1

1.1 History of Aviation . . . 2

1.1.1 Pre-Wright Era: Early Aviation . . . 2

1.1.2 The 19th Century . . . 2

1.1.3 Era of Strut-and-Wire Biplanes: 1900 to World War I . . . 3

1.1.4 Before World War II . . . 3

1.1.5 Era of Propeller-Driven Airplane: During World War II . . . 4

1.1.6 Era of Jet-Propelled Airplane: After World War II until end of 20th Century . . . 4

1.2 What do we study in aircraft structure? . . . 4

1.2.1 Design Philosophy . . . 5

1.2.2 Development of Aircraft Structures . . . 5

1.3 Loads Acting on an Aircraft . . . 6

1.4 Rotations Acting on an Airplane . . . 7

1.5 Components of a typical Aircraft . . . 8

1.5.1 Wings . . . 10

1.5.2 Fuselage . . . 16

1.5.3 Horizontal stabilizer and Elevators . . . 17

1.5.4 Stabilator . . . 18

1.5.5 Vertical Stabilizer and Rudder . . . 18

1.5.6 Spoilers . . . 19

1.5.7 Ailerons . . . 21

1.5.8 Flaps and Slats . . . 21

1.5.9 Gas Turbine Engines . . . 24

1.5.10 Landing gear . . . 25

1.6 Basic Structural Elements . . . 29

1.6.1 Wing Structure . . . 29 1.6.2 Fuselage Structure . . . 31 1.6.3 Semimonocoque Structures . . . 33 1.7 Materials . . . 37 1.8 References . . . 38 1.9 Suggested Problems . . . 39

Chapter 2. Principle of Aerodynamics 40 2.1 Aerodynamics . . . 41

2.1.1 Continuity . . . 41

2.1.2 Newton’s Laws Of Motion . . . 41

2.1.3 Conservation laws . . . 43

(13)

2.2 Mach Number . . . 47 2.3 Dynamic pressure . . . 48 2.4 Aircraft Weight . . . 49 2.5 Center of Gravity . . . 50 2.6 Center of Pressure . . . 51 2.7 Aerodynamic Center . . . 53 2.8 Lift . . . 53

2.8.1 How is lift generated . . . 53

2.8.2 No Fluid, No Lift . . . 53

2.8.3 No Motion, No Lift . . . 54

2.8.4 Factors That Affect Lift . . . 54

2.8.5 Lift Equation . . . 54

2.9 Drag . . . 55

2.10 Normal Load Factor . . . 56

2.10.1 Equations of Motion . . . 56

2.10.2 Steady, Level Flight . . . 57

2.10.3 Level Turn: Pull-Up . . . 58

2.10.4 Level Turn: Pull-Down . . . 58

2.10.5 Banked Turns . . . 58

2.11 References . . . 60

2.12 Suggested Problems . . . 61

Chapter 3. Load Analysis 62 3.1 Newton’s Laws . . . 62

3.2 Units . . . 63

3.2.1 Importance of Units . . . 63

3.2.2 Systems of Units . . . 64

3.3 Load Analysis . . . 68

3.3.1 Internal Force Sign Convention . . . 68

3.4 Load Diagrams . . . 77

3.4.1 Sign Conventions . . . 77

3.4.2 Linear Differential Equations of Equilibrium . . . 80

3.5 Discrete Load Diagrams . . . 93

Chapter 4. Thin-Wall Cross-Sectional Properties 110 4.1 Geometric Properties of Plane Areas . . . 111

4.1.1 Area . . . 111

4.1.2 First Moments of Area . . . 111

4.1.3 Centroid of an Area . . . 113

4.1.4 Second Moments of Area . . . 113

4.1.5 Polar Moment of Inertia . . . 115

4.1.6 Radius of Gyration . . . 115

4.2 Modulus-Weighted Properties of Plane Areas . . . 125

4.2.1 Area . . . 125

(14)

4.3 Properties of Plane Areas of Thin-Walls . . . 138

4.3.1 Area . . . 138

4.3.2 First Moments of Area . . . 139

Chapter 5. Applied Linear Elasticity 193 5.1 Theory of Stresses . . . 194

5.1.1 State of Stress at a Point . . . 194

5.1.2 Stress Convention and Signs . . . 198

5.1.3 Equilibrium . . . 199

5.1.4 Surface Equilibrium: Cauchy’s Stress Relation . . . 205

5.1.5 Principal Stresses and Principal Planes . . . 213

5.2 State of Plane Stress . . . 225

5.2.1 Principal stresses for Plane State of Stress . . . 226

5.2.2 Principal stresses: Eigenvalue Approach . . . 226

5.2.3 Principal stresses: Transformation Equations Approach . . . 227

5.2.4 Principal stresses: Mohr’s Circle Approach . . . 228

5.3 Important Stresses . . . 252

5.3.1 Octahedral Stresses . . . 252

5.3.2 Von Mises Stress . . . 255

5.4 Theory of Strains . . . 257

5.4.1 State of Strain . . . 260

5.4.2 Strain compatibility equations . . . 268

5.4.3 Cauchy’s relationship for Strains . . . 270

5.4.4 Principal Strains and Principal Planes . . . 273

5.5 State of Plane Strain . . . 283

5.5.1 Principal strains for State of Plane Strain . . . 283

5.5.2 Strain Measurements . . . 285

5.6 Alternative Stress and Strain Quantities . . . 291

5.6.1 Green-Lagrange strains . . . 291

5.6.2 Stress Measures . . . 296

(15)

Chapter 6. Mechanical Behavior of Aerospace Materials 307

6.1 Constitutive Equations for Elastic Materials . . . 308

6.1.1 Hooke’s Law . . . 308

6.1.2 Internal Strain Energy . . . 309

6.1.3 Anisotropic Materials . . . 310

6.1.4 Elastic Constitutive Relationship for Isotropic Materials . . . 315

6.1.5 Elastic Stress-Strain Relationship for Orthotropic Materials . . . 318

6.1.6 Temperature Strains in Isotropic Materials . . . 319

6.2 Plane Stress and Plane Strain . . . 321

6.2.1 Consequence of Plane Stress . . . 321

6.2.2 Consequence of Plane strain . . . 322

6.2.3 von Mises Stress in Plane Strain and Plane Stress . . . 322

Chapter 7. Advanced Beam Theories 382 7.1 Beam Theory . . . 383

7.1.1 Basic Considerations . . . 383

7.1.2 Principle of Saint-Venant . . . 383

7.1.3 Internal Force Sign Convention . . . 384

7.1.4 Resultant Forces and Moments . . . 385

7.2 Euler-Bernoulli Beam Theory . . . 386

7.2.1 Displacement Field . . . 388

7.2.2 Curvatures . . . 389

7.2.3 Strains-displacement Equations . . . 391

7.2.4 Stress-Strain Equations . . . 394

7.2.5 Neutral Axis . . . 395

7.2.6 Axial Stresses for Linear Thermoelastic Heterogeneous Beams . . . 395

7.2.7 Equations of Equilibrium . . . 400

7.2.8 Slope and deflection diagrams . . . 401

7.3 Timoshenko Beam Theory . . . 414

7.3.1 Displacement Field . . . 414

7.3.3 Stress-Strain Equations . . . 416

7.3.4 Axial Stresses for Linear Thermoelastic Heterogeneous Beams . . . 417

7.3.5 Slope and deflection diagrams . . . 422

7.4 Plane Stress: Thick Beams . . . 435

7.4.1 Plane Stress . . . 435

7.4.2 Stress-Strain relationship . . . 436

7.4.3 Compatibility Equations . . . 436

7.4.4 Equilibrium Equations . . . 437

7.4.5 Plane Stress Elasticity Problem . . . 437

7.4.6 Plane Stress Elasticity Solution via Airy Stress Function . . . 438

7.5 Classical (St. Venant0_{s) Torsion Theory . . . 472}

7.5.1 Displacement field . . . 473

(16)

7.5.4 Equilibrium Equations . . . 476

7.5.5 Boundary Conditions . . . 477

7.5.6 Alternative Procedure . . . 478

Chapter 8. Thin-Walled Beam Analysis 494 8.1 Thin-walled Beam Shear in Open Sections . . . 495

8.1.1 No thermal loads . . . 502

8.1.2 No axial and thermal loads . . . 502

8.1.3 No axial and thermal loads and symmetric . . . 502

8.1.4 No axial, distributed, and thermal loads and symmetric . . . 502

8.1.5 Procedure to Calculate the Shear Flow in Open Sections . . . 503

8.1.6 Shear Flow in Multiweb Junctions . . . 512

8.2 Shear Center in Thin-Walled Open Sections . . . 521

8.2.1 Definition of Shear Center . . . 521

8.2.2 Static Equivalence . . . 522

8.2.3 General Procedure . . . 523

8.3 Torsion in Open Thin-Walled Sections . . . 531

8.3.1 Prandtl’s membrane analogy for torsion . . . 531

8.3.2 Torsion of a Narrow Rectangular Cross-Section . . . 534

8.3.3 Torsion of an Arbitrary Open Thin-walled Cross-Sections . . . 536

8.4 Cross-section Idealization . . . 544

8.4.1 Idealization of Semi-Monocoque construction . . . 544

8.4.2 Typical method to Idealize of webs . . . 545

8.4.3 Shear flow and shear center in Open Idealized Sections . . . 548

8.5 Closed Single-Cell Thin-Walled Sections . . . 561

8.5.1 Enclosed area . . . 561

8.5.2 Bredt’s formula . . . 566

8.5.3 Shear flow due to transverse shear . . . 584

8.5.4 Solution procedure to obtain shear center in closed sections . . . 585

8.6 Analysis of Thin-walled Multi-Cell Closed Sections . . . 601

8.6.1 Bending . . . 601

8.6.2 Pure Torsion . . . 601

8.6.3 Pure Shear . . . 609

8.7 Analysis of Combined Open and Closed Thin-walled sections . . . 610

8.7.1 Bending . . . 610

8.7.2 Pure Shear . . . 610

8.7.3 Pure Torsion . . . 611

Chapter 9. Virtual Work Principles 623 9.1 Differential Work and Virtual Work . . . 624

9.1.1 Differential Work . . . 624

9.1.2 Virtual Work . . . 625

(17)

9.2 Review of equations of linear elasticity . . . 626

9.3 PVW for a System of Particles . . . 628

9.3.1 Virtual Displacements . . . 628

9.3.2 PVW of a particle . . . 630

9.3.3 PVW for rigid and deformable bodies . . . 631

9.3.4 Procedure . . . 633

9.4 PVW for Deformable Continuous Structures . . . 648

9.4.1 PVW for an Elastic Bar . . . 652

9.4.2 PVW for an Elastic Truss Bar . . . 661

9.4.3 Stiffness Influence Coefficients . . . 667

9.4.4 Procedure . . . 668

9.4.5 Strain Energy: Castigliano’s First Theorem . . . 688

9.4.6 PVW for Beams . . . 696

9.5 Principle of Complementary Virtual Work . . . 716

9.6 PCVW for a System of Particles . . . 717

9.7 PCVW for Continuous Deformable Bodies . . . 722

9.7.1 PCVW for a Bar . . . 723

9.7.2 PCVW for a Truss . . . 730

9.7.3 Procedure . . . 733

9.7.4 Complementary Strain Energy: Castigliano’s Second Theorem . . . 744

9.7.5 PCVW for a Beam . . . 752

9.7.6 PCVW for Frames . . . 772

Chapter 10. Failure Theories for Static Loading 784 10.1 Uncertainties in Design . . . 785

10.1.1 Design Safety factors . . . 785

10.1.2 Margin of Safety . . . 790

10.2 Ductile and Brittle Failure Theories . . . 792

10.3 3-D Stress State Failure Theories: Brittle Materials . . . 794

10.3.1 Maximum Normal Stress Criterion . . . 794

10.3.2 Brittle Coulomb-Mohr Criterion . . . 795

10.3.3 Comparison of MNS and BCM Criterions . . . 796

10.4 3-D Stress State Failure Theories: Ductile Materials . . . 797

10.4.1 Aka Distortion Energy Criterion . . . 798

10.4.2 Maximum Shear Stress Criterion . . . 801

10.4.3 Comparison of DE and MSS Criterions . . . 805

10.4.4 Ductile Coulomb-Mohr Criterion . . . 806

10.5 Introduction to Fracture Mechanics . . . 820

10.5.1 Fracture of Cracked Members . . . 821

10.5.2 Cracks as stress raisers . . . 821

10.5.3 Fracture toughness . . . 823

10.5.4 Fracture Mechanics: MODE I . . . 824

10.5.5 Fracture Mechanics: Tables and Plots . . . 828

10.5.6 Fracture Mechanics: Mixed Modes . . . 832

(18)

10.5.8 Plastic zone . . . 835

10.5.9 Plane stress . . . 836

10.5.10 Plane strain . . . 836

10.5.11 Plasticity limitations on LEFM . . . 836

10.5.12 Fracture toughness in plane strain and plane stress . . . 838

10.5.13 Superposition of Combined Loading . . . 846

Chapter 11. Failure Theories for Dynamic Loading 863 11.1 Vibration Analysis . . . 864

11.1.1 Fundamental Natural Frequency . . . 864

11.2 Impact . . . 870

11.2.1 Assumptions . . . 870

11.2.2 Freely falling body . . . 870

11.2.3 Falling body with a velocity . . . 872

11.2.4 Horizontally Moving Weight . . . 873

11.2.5 Maximum Dynamic Load and Stress . . . 873

11.3 Fatigue . . . 880 11.3.1 Cyclic Stresses . . . 880 11.3.2 Fluctuating . . . 881 11.3.3 Fully Reversed . . . 882 11.3.4 Repeated (Tension) . . . 883 11.3.5 Repeated (Compression) . . . 883

11.4 Alternate and mean stresses . . . 884

11.4.1 Ductile materials . . . 884

11.4.2 Brittle materials . . . 885

11.5 S–N Diagrams . . . 885

11.5.1 Fatigue Regimens . . . 886

11.5.2 Endurance Stress . . . 886

11.5.3 Modified Endurance Stress . . . 887

11.5.4 Stress concentration factor . . . 889

11.5.5 Plotting S-N Diagrams . . . 891

11.5.6 Fatigue Theories of Fatigue Failure . . . 893

11.6 Cumulative fatigue damage . . . 917

Chapter 12. Structural Stability 935 12.1 Concept of Stability of Equilibrium . . . 936

12.1.1 Buckling . . . 936

12.1.2 Stability of equilibrium . . . 936

12.1.3 Various Equilibrium Configurations . . . 938

12.1.4 Methods of stability analysis . . . 939

12.2 Stability of Rigid Bars . . . 940

12.2.1 Analysis of a Perfect System . . . 940

(19)

12.3 Stability of Beam-Columns . . . 961

12.3.1 Perfect Beam-Columns (Adjacent Equilibrium Method) . . . 961

12.3.2 Several Type of Column End Constraint . . . 975

12.3.3 Imperfect Beam-Columns (Adjacent Equilibrium Method) . . . 986

12.3.4 Perfect Beam-Column (Energy Approach) . . . 992

12.3.5 Inelastic Buckling . . . 1002

12.4 References . . . 1003

Chapter 13. Introduction to Aeroelasticity 1014 13.1 Definitions . . . 1015

13.2 Static Aeroelasticity . . . 1019

13.2.1 Divergence Analysis of A Rigid Wing . . . 1019

13.2.2 Divergence Analysis of Flexible Straight Wings . . . 1023

13.2.3 Divergence Analysis of Flexible Swept Wings . . . 1028

13.2.4 Aileron Reversal speed . . . 1033

13.3 The flight envelops . . . 1034

13.3.1 Basic Maneuver V –n Diagram: No gust loads . . . 1037

13.3.2 Wing Design . . . 1045

13.3.3 Design Gust Load Factors . . . 1048

13.4 References . . . 1055

Appendix A. Math Review Using MatLab 1060 A.1 What is MATLABr _{. . . 1061}

A.1.1 Getting Familiar with MATLABr _{. . . 1061}

A.1.2 Basic commands and syntax . . . 1062

A.1.3 MATLABr _{Help command . . . 1063}

A.1.4 M-Files . . . 1065

A.1.5 Programming in MATLABr _{. . . 1067}

A.1.6 Diary on and diary off . . . 1071

A.1.7 Graphical Display of Functions . . . 1072

A.1.8 Final Remarks on MATLABr _{. . . 1077}

A.2 Linear Algebra . . . 1078

A.2.1 Matrices . . . 1078

A.2.2 Vectors . . . 1086

A.2.3 Matrix and Vector Operations . . . 1091

A.2.4 General Rules for Matrix Operations . . . 1117

A.2.5 Norm of a Vector . . . 1118

A.3 Solution to Linear System of Equations . . . 1121

A.4 Polynomial Approximation . . . 1125

A.4.1 Lagrange Interpolation Functions . . . 1125

A.4.2 Newton Interpolating Polynomial . . . 1126

A.4.3 Hermite Interpolation Polynomial . . . 1127

A.5 Roots of polynomials . . . 1128

A.5.1 Linear Equations . . . 1128

(20)

A.5.3 Cubic Equations . . . 1128

A.6 Eigenvalue Problem . . . 1132

A.7 References . . . 1138

A.8 Suggested Problems . . . 1139

Appendix B. Overview Mohr’s Circle 1144 B.1 Mohr’s Circle in Stress Analysis . . . 1144

B.2 Procedure for the Mohr’s Circle . . . 1147

B.3 Mohr’s Circle in Three-Dimensional Stresses . . . 1151

B.4 Final Remarks . . . 1158

B.5 References . . . 1158

B.6 Suggested Problems . . . 1160 Appendix C. Strain-Gradient Matrix Expressions 1161

(21)

Figure 1.1 Typical forces acting on an airplane.. . . 6

Figure 1.2 Airplane rotations and body axes.. . . 8

Figure 1.3 Airplane parts (in blue) and their functions (in red).. . . 9

Figure 1.4 Geometry and nomenclature of a wing.. . . 10

Figure 1.5 The angle of attack is the angle between the chord of the airfoil and the relative wind.. 15

Figure 1.6 Lift versus the angle of attack.. . . 16

Figure 1.7 Body of an airplane.. . . 16

Figure 1.8 Horizontal stabilizer and elevator of an airplane.. . . 17

Figure 1.9 Stabilator of a fighter aircraft.. . . 18

Figure 1.10 Vertical stabilizer and rudder of an airplane.. . . 19

Figure 1.11 Spoilers of an airplane.. . . 20

Figure 1.12 Ailerons of an airplane.. . . 21

Figure 1.13 Flaps and slats of an airplane.. . . 22

Figure 1.14 Flaps partially deployed (left), full flaps (middle), full flaps with spoilers deployed (right).. . . 23

Figure 1.15 The position of the leading edge slats on an airliner (Airbus A300).. . . 23

Figure 1.16 Gas turbine engines on various aircraft.. . . 24

Figure 1.17 The main undercarriage and nose undercarriage of a Qatar Airways A330-300 (A7-ACA).. . . 25

Figure 1.18 The main undercarriage and nose undercarriage of a Qatar Airways A330-300 (A7-ACA).. . . 25

Figure 1.19 Wing and fuselage undercarriages on a Boeing 747.. . . 26

Figure 1.20 Landing gear parts of a Boeing 737-700.. . . 27

Figure 1.21 Different angles of a Boeing 757 landing gear (12 o’clock, 10 o’clock, 3 o’clock, 4 o’clock, 6 o’clock, 9 o’clock, 11 o’clock).. . . 28

Figure 1.22 Spar construction.. . . 30

Figure 1.23 Typical spar construction.. . . 31

Figure 1.24 (a) Spars only, (b) spars and stringers.. . . 31

Figure 1.25 Wing cross-sections with integrally stiffened skin.. . . 32

Figure 1.26 Fuselage structure.. . . 32

Figure 1.27 Monocoque and semi-monocoque structure.. . . 34

Figure 1.28 Typical semimonocoque aircraft structures.. . . 35

Figure 1.29 Idealization of semimonocoque structure: (a) actual structure, (b) idealized structure.. 36

Figure 1.30 Idealization of monocoque shell: (a) actual structure, (b) idealized structure.. . . 36

Figure 2.1 Newton’s third law applied to aerodynamics.. . . 43

Figure 2.2 Aircraft weight.. . . 49

Figure 2.3 Center of pressure.. . . 51

Figure 2.4 Forces acting on an aircraft. . . 57

Figure 2.5 Forces acting on an aircraft during a vertical pull-up. . . 58

Figure 2.6 Forces acting on an aircraft during a vertical pull-down. . . 59

Figure 2.7 Forces acting on an aircraft during a banked turn. . . 60

(22)

Figure 3.1 Positive sign convention.. . . 68 Figure 3.2 Three-dimensional bar-structure.. . . 69 Figure 3.3 Free body diagrams for the three-dimensional bar-structure.. . . 70 Figure 3.4 Equilibrium element supporting a general force system under the stress convention

in the x-y plane.. . . 77 Figure 3.5 Equilibrium element supporting a general force system under the structural

conven-tion in the x-y plane. . . 78 Figure 3.6 Equilibrium element supporting a general force system under the elasticity

conven-tion in the x-y plane. . . 79 Figure 3.7 Machine component for example below.. . . 81 Figure 3.8 Dimensionless axial load distribution.. . . 85 Figure 3.9 Dimensionless shear (in the y–axis) load distribution.. . . 85 Figure 3.10 Dimensionless shear (in the z–axis) load distribution.. . . 86 Figure 3.11 Dimensionless torsional load distribution.. . . 86 Figure 3.12 Dimensionless moment (about the y–axis) distribution.. . . 87 Figure 3.13 Dimensionless moment (about the z–axis) distribution.. . . 87 Figure 3.14 Cross-section of the helicopter blade.. . . 88 Figure 3.15 Loading on the helicopter blade.. . . 88 Figure 3.16 Replacing the pressure g(x, z) with a distributed load py(x).. . . 89

Figure 3.17 Locating all loads at the weighted-modulus centroid.. . . 90 Figure 4.1 Arbitrary cross section of a structure.. . . 111 Figure 4.2 Arbitrary cross section of a structure.. . . 112 Figure 4.3 Typical cross section of a thin-walled structure.. . . 138 Figure 5.1 Solid body in equilibrium.. . . 194 Figure 5.2 Solid body in equilibrium sliced with an arbitrary plane.. . . 195 Figure 5.3 Complete definition of the state of stress at a point.. . . 199 Figure 5.4 Shear stresses on the faces of an element at a point in an elastic body about the

z-axis.. . . 200 Figure 5.5 Shear forces on the faces of an element at a point in an elastic body about the z-axis..203 Figure 5.6 This is an infinitesimal element representing the state of stress for the given problem

(NOTE: Units are part of the answer). . . 206 Figure 5.7 Principal state of stress. . . 217 Figure 5.8 Positive stresses on a two dimensional element.. . . 225 Figure 5.9 a) Stresses acting on an element in plane stress. b) Stresses acting on an element

oriented at an angle θ = α. c) Principal normal stresses. d) Maximum in-plane shear stresses..231 Figure 5.10 Mohr’s circle for plane stress in the y-z plane. . . 233 Figure 5.11 a) Stresses acting on an element in plane stress. b) Stresses acting on an element

oriented at an angle θ = α. c) Principal normal stresses. d) Maximum in-plane shear stresses..236 Figure 5.12 Mohr’s circle for plane stress in the x-z plane. . . 238 Figure 5.13 a) Stresses acting on an element in plane stress. b) Stresses acting on an element

oriented at an angle θ = α. c) Principal normal stresses. d) Maximum in-plane shear stresses..240 Figure 5.14 Mohr’s circle case for uniaxial state of stress. . . 242 Figure 5.15 Mohr’s circle case for triaxial state of stress. . . 244 Figure 5.16 Mohr’s circle case for hydrostatic state of stress. . . 246 Figure 5.17 General state of stress for stresses acting on octahedral planes. . . 252 Figure 5.18 Tetrahedron element at O. . . 253

(23)

Figure 5.19 Deformation of a solid body from the initial configuration, C0_{, to the current}

config-uration, C1_{. . . 257}

Figure 5.20 The neighborhood of point P in the reference and deformed configurations... . . 258 Figure 5.21 Shear deformation in the reference and deformed configurations... . . 262 Figure 5.22 Three strain gauges at the surface of a solid: 3-gage rosette.. . . 285 Figure 5.23 Mohr’s circle for plane strain in the x-y plane. . . 288 Figure 6.1 Uniaxial loading-unloading stress-strain curves. . . 308 Figure 6.2 Strain energy density.. . . 310 Figure 6.3 Mohr’s circle case for the principal state of stress. . . 329 Figure 7.1 Sign convention for stress resultants on a beam cross section.. . . 385 Figure 7.2 Stresses acting on a beam’s cross-sectional differential volume.. . . 386 Figure 7.3 Decomposition of the axial displacement field.. . . 388 Figure 7.4 Definition of curvature. . . 389 Figure 7.5 Bending of a beam element of length dx in the x-y plane.. . . 392 Figure 7.6 Cross-section of the helicopter blade.. . . 407 Figure 7.7 Shear deformation of a beam element about the z-axis.. . . 414 Figure 7.8 Cross-section of the helicopter blade.. . . 428 Figure 7.9 Cantilever rectangular beam carrying a point load at the tip.. . . 441 Figure 7.10 Cantilever rectangular beam carrying a uniform shear load on upper surface.. . . 451 Figure 7.11 Simply-supported rectangular beam carrying a uniform normal load on upper surface..460 Figure 7.12 Cylindrical bar of arbitrary cross-section in pure torsion.. . . 472 Figure 7.13 Cylindrical bar of arbitrary cross-section in pure torsion.. . . 473 Figure 7.14 Representation of stress state along edge of solid cross-section under torsion.. . . 479 Figure 7.15 Representation of stress state at top cross-section of rod under torsion.. . . 480 Figure 7.16 Linear elastic torsion of a shaft with a rectangular cross-section.. . . 492 Figure 7.17 Linear elastic torsion of a shaft with a parabolic cross-section.. . . 493 Figure 8.1 Open thin-walled section.. . . 495 Figure 8.2 Shear stress in open thin-walled section.. . . 496 Figure 8.3 Shear flow in arbitrary open thin-walled section.. . . 497 Figure 8.4 Differential element of a thin-walled beam showing shear flow and bending stress.. . 498 Figure 8.5 Shear flow convention in thin-walled open channel section.. . . 503 Figure 8.6 Unsymmetrically thin-walled channel section.. . . 505 Figure 8.7 Shear flow convention for the given thin-walled channel section.. . . 506 Figure 8.8 Shear flow on a differential portion of a multiweb junction.. . . 512 Figure 8.9 Shear flow convention.. . . 515 Figure 8.10 Tip-loaded cantilever beam: twisting and bending (first & two), and bending only

(third).. . . 521 Figure 8.11 Vertical and horizontal applied shear forces.. . . 522 Figure 8.12 Location of the shear center of a thin-walled open section.. . . 523 Figure 8.13 Unsymmetrically thin-walled channel section.. . . 524 Figure 8.14 Suggested shear flow convention for statically equivalence by taking the torque at

point 3.. . . 525 Figure 8.15 Suggested shear flow convention for statically equivalence by taking the torque at

point 2.. . . 525 Figure 8.16 Shear flow convention.. . . 529 Figure 8.17 Membrane analogy: in-plane and transverse loading.. . . 531

(24)

Figure 8.18 Equilibrium of element of a membrane.. . . 532 Figure 8.19 Torsion of a narrow rectangular strip.. . . 534 Figure 8.20 Representation of cross-section for membrane analogy and the side-view of the

mem-brane under pressure.. . . 534 Figure 8.21 Arbitrary open thin-walled cross-Section.. . . 536 Figure 8.22 Actual thin-walled section and idealized section.. . . 544 Figure 8.23 Actual thin-walled section and idealized section.. . . 545 Figure 8.24 Idealization of a thin rectangular wall into two concentrated areas.. . . 545 Figure 8.25 Idealization of a thin rectangular wall into two concentrated areas.. . . 547 Figure 8.26 Actual thin-walled section and idealized section.. . . 548 Figure 8.27 Symmetrically thin-walled channel section.. . . 551 Figure 8.28 Shear flow convention. . . 552 Figure 8.29 Shear flow convention for statically equivalence when taking the torque at point 2.. . 555 Figure 8.30 Shear flow convention for statically equivalence when taking the torque at point 3. . 556 Figure 8.31 Area enclosed by the contour.. . . 561 Figure 8.32 Thin-walled, single cell beam with an arbitrary cross-sectional contour.. . . 566 Figure 8.33 Thin-walled element in its undeformed and deformed configurations.. . . 567 Figure 8.34 Geometry of the contour.. . . 569 Figure 8.35 Superposition of shear flows: problem consisting of an open section and a section

with a constant shear flow.. . . 584 Figure 8.36 Symmetrical thin-walled monocoque closed section.. . . 586 Figure 8.37 Shear flow convention. . . 590 Figure 8.38 Shear flow convention. . . 591 Figure 8.39 Shear flow convention for statically equivalence when taking the torque at point O.. 593 Figure 8.40 Shear flow convention for statically equivalence when taking the torque at point O. . 594 Figure 8.41 Shear flow convention for statically equivalence when taking the torque at point O. . 599 Figure 8.42 Multicell thin-walled beam.. . . 601 Figure 8.43 Shear flow in a Multicell thin-walled beam cross-section.. . . 602 Figure 8.44 A two-cell thin-walled section under torsion.. . . 604 Figure 8.45 A typical hybrid thin-walled wing section.. . . 610 Figure 8.46 A hybrid thin-walled section under torsion.. . . 611 Figure 8.47 Hybrid thin-walled wing section.. . . 613 Figure 9.1 Force vector and displacement vector at a location s.. . . 624 Figure 9.2 Differential work done.. . . 625 Figure 9.3 Virtual work done.. . . 625 Figure 9.4 Complementary Virtual work done.. . . 626 Figure 9.5 Virtual work done.. . . 629 Figure 9.6 Particle in equilibrium subject to n forces.. . . 630 Figure 9.7 System of particles showing both external and internal forces.. . . 632 Figure 9.8 Rigid-bars configuration for Example 9.1.. . . 636 Figure 9.9 Rigid-bars configuration for Example 9.2.. . . 640 Figure 9.10 Rigid-bars configuration for Example 9.3.. . . 644 Figure 9.11 Elastic bar subject to a load P undergoing a virtual displacement.. . . 652 Figure 9.12 Point force acting on a clamped bar. . . 655 Figure 9.13 Elastic truss bar.. . . 661 Figure 9.14 Undeformed and deformed states of the qth elastic truss bar.. . . 662

(25)

Figure 9.15 Planar truss configuration.. . . 668 Figure 9.16 Idealized landing gear truss structure.. . . 670 Figure 9.17 Four bar truss structure.. . . 673 Figure 9.18 Four bar truss structure.. . . 678 Figure 9.19 Representation of a single bay of a wing spar truss.. . . 683 Figure 9.20 Virtual strain energy density per unit volume and virtual complementary strain

energy density per unit volume. . . 688 Figure 9.21 Four bar truss structure.. . . 692 Figure 9.22 Distributed load acting on a clamped beam. . . 698 Figure 9.23 Distributed load acting on a simply-supported beam. . . 707 Figure 9.24 Nondimensional axial displacement with one and five term approximation. . . 714 Figure 9.25 Nondimensional transverse displacement with one and five term approximation. . . . 714 Figure 9.26 Nondimensional lateral displacement with one and five term approximation. . . 715 Figure 9.27 System in its deformed and undeformed state. . . 717 Figure 9.28 Virtual loads acting on the system of particles. . . 718 Figure 9.29 Two rigid pin-connected members joined by a spring. . . 719 Figure 9.30 Elastic bar subject to a virtual load δP .. . . 723 Figure 9.31 Uniform, homogeneous, elastic bar subject to an axially distributed load. . . 727 Figure 9.32 Elastic truss bar.. . . 730 Figure 9.33 Planar truss configuration.. . . 733 Figure 9.34 Truss structure. . . 735 Figure 9.35 Truss structure. . . 739 Figure 9.36 Virtual strain energy density per unit volume and virtual complementary strain

energy density per unit volume. . . 744 Figure 9.37 Truss structure. . . 747 Figure 9.38 Distributed load acting on a simply-supported beam. . . 754 Figure 9.39 Distributed load acting on a simply-supported beam. . . 759 Figure 9.40 Distributed load acting on a simply-supported beam. . . 772 Figure 9.41 Truss bar structure. . . 779 Figure 9.42 Strut-braced wing subjected to a point load P . . . 780 Figure 9.43 Distributed load acting on a simply-supported beam. . . 781 Figure 9.44 Elastic circular arch supporting a load P . . . 782 Figure 9.45 Idealized truss-bar structure supporting a load P . . . 783 Figure 10.1 Three modes of fracture. . . 820 Figure 11.1 Typical S-N diagram for ferrous materials.. . . 886 Figure 12.1 Equilibrium states. . . 938 Figure 12.2 One degree of freedom structural configuration, α = 0. . . 940 Figure 12.3 Summary of the primary and secondary path stability. . . 947 Figure 12.4 One degree of freedom structural configuration, α > 0. . . 949 Figure 12.5 Linear response for various levels of imperfection. . . 950 Figure 12.6 Nonlinear response for various levels of imperfection. . . 951 Figure 12.7 One degree of freedom structural configuration. . . 952 Figure 12.8 Summary of the primary and secondary path stability. . . 959 Figure 12.9 Load and frequency plots against vertical deflection. . . 960 Figure 12.10 A simply-supported beam column subject to an axial load. . . 964 Figure 12.11 Cantilevered beam column subject to an axial load. . . 967

(26)

Figure 12.12 A clamped-spring supported beam column subject to an axial load. . . 970 Figure 12.13 Response for various levels of load imperfection. . . 989 Figure 12.14 Response for various levels of geometric imperfection. . . 991 Figure 12.15 A simply-supported beam column subject to an axial load. . . 995 Figure 12.16 Beam column with unsymmetrical supports subject to an axial load. . . 999 Figure 12.17 Configuration case A.. . . 1006 Figure 12.18 Configuration case B.. . . 1007 Figure 12.19 Configuration case C.. . . 1008 Figure 12.20 Rigid bar with a concentrated mass and spring system.. . . 1009 Figure 12.21 A spring-supported beam column subject to an axial load. . . 1011 Figure 12.22 A simply-supported beam column subject to an axial load. . . 1012 Figure 13.1 Interdisciplinary nature of the field of aeroelasticity. . . 1015 Figure 13.2 The aeroelastic triangle of loads. . . 1016 Figure 13.3 A two-dimensional rigid wing model to study divergence. . . 1019 Figure 13.4 The divergence dynamic pressure with respect the angle of attack. . . 1021 Figure 13.5 Slender straight wing subject to distributed torsional load. . . 1023 Figure 13.6 Small element with the differential loads acting on the wing. . . 1024 Figure 13.7 Top view of an aircraft with swept wings. . . 1028 Figure 13.8 Reverse airflow: forward-swept wing vs. aft swept wing.. . . 1030 Figure 13.9 Aerodynamic axes on a two-dimensional model.. . . 1038 Figure 13.10 Lift coefficient vs. angle of attack.. . . 1039 Figure 13.11 The V –n diagram for a typical aircraft. . . 1040 Figure 13.12 Maneuver V –n diagram. . . 1043 Figure 13.13 Aircraft subject to gust loads. . . 1048 Figure 13.14 Aircraft subject to gust loads. . . 1050 Figure A.1 Basic MATLABr _{working environment.. . . 1063}

Figure B.1 Typical Mohr’s circles for a given state of stress.. . . 1145 Figure B.2 Sketch of the given information on the Mohr’s circle.. . . 1148 Figure B.3 a) Stresses acting on an element in plane stress. b) Stresses acting on an element

oriented at an angle θ = α. c) Principal normal stresses. d) Maximum in-plane shear stresses..1151 Figure B.4 Mohr’s circle for plane stress in the x-y plane. . . 1154 Figure B.5 a) Stresses acting on an element in plane stress. b) Stresses acting on an element

(27)

Table 7.1 Shear constant for various cross-sections (Shames and Dym).. . . 420 Table 10.1 Semiquantitative assessment of rating factors. . . 790 Table 10.2 Plane strain fracture toughness and corresponding tensile properties for representative metals

at room temperature. “Mechanical Behavior of Materials”, by N.E. Dowling, Prentice-Hall Inc, NJ, 1999. Page 291. . . 828

Table 12.1 Effective length coefficient CL for several type of column end constraints . . . 975

(28)

Learning about Aircraft Structures

Instructional Objectives of Chapter 1

After completing this chapter, the reader should be able to:

1. Understand how aerospace structures have evolved with history. 2. Identify the main aerodynamic loads acting on airplanes.

3. Explain the major structural components of an aircraft and understand their func-tion(s).

4. Understand the use of semi-monocoque structures in modern aircraft.

Before we begin our journey to study aircraft structures let us understand what we mean by aircraft structures. We can define aircraft structures as the study of methods for designing and manufacturing aircraft, and ensuring they withstand any stress or strain. Although we could spend several books on the subject, for this introductory study we will limit our study to the few major structural components of an aircraft such as wing ribs, stringers or longerons, spars, heavy frames and bulkheads, skin, and truss components. These structural components play an important role most aircraft’s structural integrity. Their location, weight, design, material, etc. are crucial for an optimum design.

In this chapter we will learn how structural components in modern aircraft have evolved. Followed by a brief explanation of the typical loads acting on an aircraft and what are roles of the various components on a typical aircraft. We conclude this chapter studying the difference of monocoque and semi-monocoque structures.

(29)

1.1 History of Aviation

Before we begin the study of aircraft structures let us review the history of aviation at a glance. This will help us better understand the why-we-do-what-we-do. We define aviation as the design, manufacture, use, or operation of aircraft. By aircraft we mean any vehicle capable of flying. Mainly, we have two kinds of aircraft: (i) heavier-than-air (airplanes, autogiros, gliders, helicopters, and ornithopters), (ii) and lighter-than-air (balloons and airships).

Man has always wanted to overcome the challenges to move through air, water and ground. When man succeeded to travel through water and on ground, he dreamed to soar with the birds. These dreams caused accidents due to structural failures. Icarus, from the Greek mythology, is famous for his death caused when the sun melt the wax holding his artificial wings together. Years later, inventors such as Leonardo Da Vinci, John Stringfellow, and Lawrence Hargrave designed intriguing flying machines long before the Wright brothers’ famous first flight at Kitty Hawk.

1.1.1 Pre-Wright Era: Early Aviation

Roughly speaking, the first type of aircraft was a kite. It was designed in China during the fifth century by Mozi and Lu Ban. These first kites were made with silk using bamboo as the framework. During the thirteenth century, Roger Bacon, famous Franciscan friars, concluded that air could support a craft just like water supports boats. Later during the sixteenth century, Leonardo da Vinci while studying the birds’ flight designed the airscrew, leading to the propeller later on and the parachute. Leonardo was a pioneer in the design of heavier-than-air crafts. Although his designs were not successful, we was the first to conclude that the human power was insufficient to generate flight. His three most important contributions are: (i) the helicopter powered by four men, (ii) the light hang glider, (iii) and the ornithopter (a machine with mechanical wings which flap to mimic a bird).

In June of 1783, in Annonay (France), the Montgolfier brothers (Joseph and Jacques), were the first to succed in launching a human to air. Their design consisted in a hot air balloon made of silk and lined with paper to trap the gas, called the Montgolfiere. This first successful flight lifted 6,562 feet into the air, traveled more than a mile and stayed aloft for about ten minutes.

The Montgolfiers believed they discovered a new gas (called Montgolfier gas) when they held a flame near the opening at the bottom, and the balloon expanded with hot air and floated upward. They thought that this gas was lighter than air and caused the inflated balloons to rise. The gas was merely air, which became more buoyant as it was heated.

1.1.2 The 19th Century

During the ninth century, new developments took place in the field of stability and trust generation. Lawrence Hargrave designed a box kite in 1893, followed by Alexander Graham Bell who experimented with box kites and wings built of multiple compound tetrahedral kites covered in silk (1907-1912). Bell named this tetrahedral kites Cygnet I, II and III. Jean Marie Le Bris designed a glider with movable

(30)

wings.

The greatest impact, during the ninth century, to the aviation industry was the integration of motors to aircraft. John Stringfellow designed a steam engine powered aircraft, Lawrence Hargrave designed a rigid-wing aircraft with flapping blades operated by compressed-air motor and the rotary engine, which powered many early aircraft up until about 1920. They realized that successful powered flight required light gasoline engines instead of the cumbersome steam previously used. Samuel Langley designed the first heavier-than air gasoline powered engine which actually flew, called the aerodrome. This aircraft was powered by a 53 horsepower 5-cylinder radial engine.

Also, during this century, British Sir George Cayley designed a combined helicopter and horizontally propelled aircraft while Francis Herbert Wenham studied the behavior of multiple wings aircraft using wind tunnels.

1.1.3 Era of Strut-and-Wire Biplanes: 1900 to World War I

The first heavier-than-air machine powered flight took place on December 17, 1903 (10:35am). It was the famous Wright Bothers, Orville and Wilbur, first flight. This flight lasted 12 second and covered a distance of 120 feet. Their mayor breakthrough was the invention of the “three axis-control”, which enabled the pilot to steer the aircraft effectively and maintain its equilibrium. This has been become the standard on fixed wing aircraft of all kinds. Unfortunately, on September 17, their aircraft crashed injuring Orville and his passenger Lieutenant Thomas E. Selfridge. Selfridge later died due to compli-cations, making him the first person to die in a powered airplane. In 1908, Wilbur completed a 2 hour and 20 minute flight, showing full control over his flyer. The flyer became the first successful military airplane and it remained in service for around two years.

On July 4, 1908, Glenn H. Curtiss flew the “June Bug” 5090 ft in 1 minute and 42.5 seconds. Curtiss achieved the following: (i) the first American award the Scientific American Trophy, (ii) win the first international speed event, (ii) and the first American to develop and fly a seaplane.

In 1913, A. V. Roe built the first tractor biplane. It consists of two-winged airplanes with engine and propeller in front of the wing. Years later, the military used the tractor biplanes with a closed fuselage as the first standard military aircraft. During the World War I, began the development of huge biplane bombers with two to four engines. Not only the military used aircraft, but the airmail also began using aircraft: on September 23, 1911 the pilot Earle Ovington completed the first airmail officially approved by the U.S. Post Office Department. Also in 1911, Calbraith P. Rogers completed the first transcontinental flight across the U.S.

1.1.4 Before World War II

During the period of 1910 to 1930, the aviation industry greatly grew. In 1919, Captain E. F. White made a nonstop flight from Chicago to New York; later in 1923, Lieutenants Oakley Kelly and John A. Macready made the first nonstop transcontinental flight from Roosevelt Field, Long Island to Rockwell Field. In 1924, Douglas World Cruiser was developed for the U.S. Army Air Service for an attempt to

(31)

make the first flight around the world.

One of the most successful designs during this period, was the Douglas DC-3 which became the first airliner to exclusively carry passengers, starting the modern era of passenger airline service. Also, mail delivery was impacted by the Kelly Air Mail act. This act authorized the postmaster general to contract for domestic airmail service with commercial air carriers. It also set airmail rates and the level of cash subsidies to be paid to companies that carried the mail. By transferring airmail operations to private companies, the government helped create the commercial aviation industry.

1.1.5 Era of Propeller-Driven Airplane: During World War II

By the beginning of World War II, many towns and cities had built airports, and there were numerous qualified pilots. The war brought many innovations to aviation such as the first jet aircraft and the first liquid-fueled rockets. The largest operator of all international airlines in operation was Pan American Airways, serving 46 countries and colonies. Before World War II, only about 193,000 people were employed in the aviation industry; and after 1941, the number increased to almost 450,000.

1.1.6 Era of Jet-Propelled Airplane: After World War II until end of 20th

Century

In August of 1939, the Heinkel He 178 became the world’s first aircraft to fly under turbojet power, thus becoming the first practical jet plane. The first operational turbojet aircraft, the Messerschmitt Me 262 and the Gloster Meteor, entered service towards the end of World War II in 1944. Civilian aircraft orders drastically increased from 6,844 in 1941 to 40,000 by the end of 1945. One of the minor military contractors was the Boeing Company who later became the largest aircraft manufacturer in the world. New aerodynamic designs, metals, and power plants would result in high-speed turbojet airplanes. These planes would later be able to fly supersonically and make transoceanic flights regularly.

During the 20th century, Burt Rutan designed hundreds of aircraft, including the now-famous Voy-ager, which was piloted by Dick, his brother, and Jeana Yeager in 1986 on a record-breaking nine-day non-stop flight around the world. The Voyager held 1,200 gallons of fuel in its 17 fuel tanks, and maintained an average speed of 115.8 mph. lasted 9 days, 3 minutes, 44 seconds and covered 25,012 miles.

1.2 What do we study in aircraft structure?

By aircraft structure we refer to study and analyze how the plane is built. An aircraft must be lightweight, but stress and strain resistant at the same time. The analysis of aircraft is quite complex, not only for its design but also due to the loads it is subject. The aircraft experiences many forces during flight and its structural components must be able to resist to these static and dynamics loads. What makes aircraft structural design unique t from other structural fields (such as buildings or ships) is that the aircraft

(32)

must be both lightweight and strong.

1.2.1 Design Philosophy

The main forces acting on an aircraft during flight are lift, drag, thrust and weight. The structural components of an aircraft affect directly or indirectly all of these four forces. However, the structure determines the aircraft’s weight. The total weight of the plane consists in: the aircraft itself (empty weight) plus the passengers, crew, baggage and freight (payload), and the fuel. This known as the takeoff weight. There must be enough lift to get the total weight of the aircraft into the air. Engineers also consider cruising weight and landing weight. These weights are the totals of the empty weight, payload weight, and the weight of the fuel at the time.

As an aircraft prepares to takeoff, we must keep the following in mind: (i) the aircraft must be able to lift the takeoff weight from the ground before the end of the runway; (ii) and the amount of fuel carried by the aircraft will depend on the traveling distance and the payload weight. Tradeoffs may have to be made. Lighter payloads for shorter runways; larger aircrafts with more fuel to carry heavy payloads long distances. Hence, weight becomes very important.

The aircraft structural design teams are responsible to design aircraft to withstand all static and dynamic (transient and suddenly applied) loads. This team should keep the following goals in mind:

1. SAFE LIFE: Consists in designing each part for minimum weight and yet assuring they will last for a long time.

2. FAIL SAFE: Consists in designing the aircraft’s components in such a coordination that if one unit fails, the other units will take on the load. In other words, the overall airframe (structure) is designed so that failure in one component doesn’t cause the whole aircraft to fall apart.

1.2.2 Development of Aircraft Structures

Early aircraft were built from very lightweight materials such as bamboo, wood, and fabric. They design was similar to bridges, with beam and truss structures. As for an example, the wings on the Wright Flyer formed a truss: the two wings used wires and bars diagonally (at an angle) to strengthen the wing against aerodynamic forces.

In general, inside the wings we had truss structures. The bars inside were called spars. The wires used on the diagonals strengthened the wing. The spars, plus the spar caps at each end, were shaped to give the wing aerodynamic features. This shape is often called the airfoil.

As technology has grown, so have the manufacturing techniques. Hence, in the early twentieth century, metal rods and pieces began to replace the wooden components. Metal skins, rolled very thin were weather-resistant as opposed to the fabric skins. The ribs and spars of the plane were made by riveting many pieces together. When aluminum alloys became available at the end of the 1920’s, ribs and spars were often stamped (cut) out of whole aluminum sheets.

(33)

At first, when the aircraft flew, the wood or metal frame took all of the stress. The fabric or thin metal skin could not withstand any of the load. Later, thicker metal skin was put on airplane frames. This thicker skin was able to share the stress. The metal frame could then be made of lighter metal. Hence, the aircraft’s final weight was lighter!

Throughout the years, with the use of metals, the basic aircraft design changed. The original biplane design (two wings) with struts and bracing wires, was no longer efficient at the higher speeds. The spars and wires caused more drag at higher speeds. By using metal skins to carry some of the load (of the frame) made the biplane design no longer necessary. Monoplanes (single wing) create much less drag than a biplane. Also, the monoplane did not have the struts and wires sticking out, like the biplane.

Nowadays, aircraft industry is moving towards developing aircraft with even materials that are even lighter such as composite materials. In fact, the design work continues in the field of aircraft structures for a better balance of weight and strength.

1.3 Loads Acting on an Aircraft

Figure 1.1: Typical forces acting on an airplane.

A force is something that produces a change in a physical quantity. It is a vector quantity and thus has both a magnitude and a direction. Figure 1.1 shows the typical forces that act on an aircraft during flight. These four forces are:

(34)

magnitude will depend on the mass of all the aircraft’s components, the amount of fuel, and any payload on board (people, baggage, freight, etc.). Although the total weight may be distributed throughout the airplane, its resultant acts through the center of gravity. In fact, we want our aircraft to rotate about its center of gravity.

2. LIFT (L): For an aircraft to fly it must overcome the total weight of the aircraft. This force is called lift. Lift is generated by the aircraft’s motion through air and its an aerodynamic force. Aerodynamic is a combination of two words aero and dynamic: aero stands for the air, and dynamic denotes motion. Hence an aerodynamic load may be defined as a load produces as an aircraft moves though air. The aerodynamics load lift is perpendicular to the flight direction and its magnitude will depend on several factors such as the shape, size, and aircraft velocity. Each aircraft part will experience certain list and the sum of these “lift forces” gives the total lift acting on the aircraft. Most of the aircraft’s lift is generated by the wings. The aircraft lift acts through a single point called the center of pressure1_.

3. DRAG (D):

During flight, there is another aerodynamic force which opposes the motion called drag. Drag acts along, but opposed, to the flight direction. As in the case of lift, many factors affect the magnitude of the drag force such as the shape of the aircraft, the “stickiness” of the air and the aircraft velocity. Just as the weight and lift, each of the of the individual components’ drags combine to produce the total aircraft drag. And like lift, drag acts through the aircraft center of pressure. 4. THRUST (T ): In order to overcome drag, an aircraft uses a propulsion system to generate

thrust. Thrust is a propulsion force and its direction depends on how the engines are attached to the aircraft. On some aircraft, such as the Harrier, the thrust direction can change to help the aircraft take off in a very short distance. The magnitude of the thrust will depends on factors associated with the propulsion system such as the type of engine, the number of engines, and the throttle setting.

For jet engines, it is often confusing to remember that aircraft thrust is a reaction to the hot gas rushing out of the nozzle. The hot gas goes out the back, but the thrust pushes towards the front. The aircraft motion will depend on the relative strength and direction of the forces shown in Fig. 1.1. If the forces are balanced, then it is said that the aircraft cruises at constant velocity. If the forces are unbalanced, the aircraft accelerates in the direction of the largest force.

1.4 Rotations Acting on an Airplane

The aircraft motion is a three-dimensional motion. Hence, we need to control the attitude or orientation of a flying aircraft in all directions. In flight, any aircraft will rotate about its center of gravity. We can define a three-dimensional coordinate system through the center of gravity with each axis of this coordinate system perpendicular to the other two axes. We can then define the orientation of the aircraft by the amount of rotation of the parts of the aircraft along these principal axes, as shown in Figure 1.2.

1_{The center of pressure is defined just like the center of gravity, but using the pressure distribution around the body} instead of the weight distribution.

(35)

Figure 1.2: Airplane rotations and body axes.

The yaw axis is perpendicular to the plane of the wings with its origin at the center of gravity and directed towards the bottom of the aircraft. A yaw motion is a movement of the nose of the aircraft from side to side. The pitch axis is perpendicular to the yaw axis and is parallel to the plane of the wings with its origin at the center of gravity and directed towards the right wing tip. A pitch motion is an up or down movement of the nose of the aircraft. The roll axis is perpendicular to the other two axes with its origin at the center of gravity, and is directed towards the nose of the aircraft. A rolling motion is an up and down movement of the wing tips of the aircraft.

1.5 Components of a typical Aircraft

Now that we have a fairly good idea about aircraft, let now us discuss its main parts and their functions. An aircraft is designed to move people and/or cargo from one place to another through air. Their shapes and sizes vary depending on the mission of the aircraft. The aircraft shown in Figure 1.3 is a turbine-powered aircraft used here to represent most civil transport aircraft.

As previously mentioned, an aircraft flies by lifting its total weight. As the aircraft moves through air, the wings generate most of the lift to hold the plane in the air; the jet engines provide the necessary