Aerospace Structures

Aerospace Structures by Eric Raymond Johnson is a 600+ page text and reference book for junior, senior, and graduate-level aerospace engineering students. The text begins with a discussion of the aerodynamic and inertia loads acting on aircraft in symmetric flight and presents a linear theory for the statics and dynamic response of thin-walled straight bars with closed and open cross-sections. Isotropic and fiber-reinforced polymer (FRP) composite materials including temperature effects are modeled with Hooke’s law. Methods of analyses are by differential equations, Castigliano’s theorems, the direct stiffness method, the finite element method, and Lagrange’s equations. There are numerous examples for the response of axial bars, beams, coplanar trusses, coplanar frames, and coplanar curved bars. Failure initiation by the von Mises yield criterion, buckling, wing divergence, fracture, and by Puck’s criterion for FRP composites are presented in the examples.

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The EPUB was released early May 2023. The EPUB contains MathML and alternative text. LaTeX sourcefiles were publicly released in October 2023. LaTeX sourcefiles are also available in Overleaf under a CC BY NC SA 4.0 license.

Stable link for this resource: https://doi.org/10.21061/AerospaceStructures

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Table of Contents

1. Function of Flight Vehicle Structural Members
2. Aircraft Loads
3. Elements of Thin-Walled Bar Theory
4. Some Aspects of the Structural Analysis
5. Work and Energy Methods
6. Applications of Castigliano's Theorems
7. Arches, Rings, and Fuselage Frames
8. Laminated Bars of Fiber-Reinforced Polymer Composites
9. Failure Initiation in FRP Compositives
10. Structural Stability of Discrete Conservative Systems
11. Buckling of Columns and Plates
12. Introduction to Aeroelasticity
13. Fracture of Cracked Members
14. Design of a Landing Strut and Wing Spar
15. Direct Stiffness Method
16. Applications of the Direct Stiffness Method
17. Finite Element Method
18. Introduction to Flexible Body Dynamics Appendix A: Linear Elasticity of Solid Bodies

Resources
Problem sets: http://hdl.handle.net/10919/104169
LaTeX sourcefiles: https://www.overleaf.com/read/skqbjscvwhjh#5decd4
PDFs (book and chapter-level) and EPUB: Available from the left side of the screen
Print (paperback - does not include the appendix): Order here

About this book
This text is evolved from lecture notes by the author for junior and senior students in the aerospace engineering curriculum at Virginia Tech. The subjects covered in the book presume some knowledge of statics, dynamics of rigid bodies, mechanics of deformable bodies, and mechanical vibrations. Several practice exercises in the text require programming, and typically the students use Mathematica1 or MATLAB 2 software to complete them. Examples in the text were programmed in Mathematica. A first semester sequence for junior students includes chapters 1 through 6. Note that chapter 3 on thinwall bar theory maybe too mathematical for some students, but can be used as a reference for the applications of the theory provided in chapter 4. The important topic of work and energy is covered in chapter 5, and chapter 6 is devoted to the application of Castigliano’s theorems to trusses, beams, and frames.

A second semester sequence for junior students includes topics selected by the instructor from chapter 7 on curved bars, and chapters 10 through 16. The influence of imperfection sensitivity on the buckling load of discrete systems is presented in chapter 10, followed by buckling of columns and plates in chapter 11. Article 11.2 is optional. Analysis for wing divergence is presented in the introduction to aeroelasticity in chapter 12. The methods of linear elastic fracture mechanics to predict critical loads for crack propagation is discussed in chapter 13. Design of a landing strut, and the optimal design of a spar subject to constraints on yielding, buckling and fracture are presented in chapter 14. Chapters 15 and 16 detail the direct stiffness method for trusses, beams and frames.

Topics appropriate for senior students are in chapters 8, 9, 17, and 18, and initial post-buckling in article 11.2. The response of closed and open section bars fabricated from a fiber-reinforced polymer composite (FRP) is presented in chapter 8, and failure initiation of FRP bars is presented in chapter 9. The finite element method applied to the extension and bending of bars is presented in chapter 17, which includes transverse shear deformations. The topic of adaptive mesh refinement in article 17.2.4 is optional. Articles 18.1 to 18.4 cover the dynamic response of lumped mass models, eigenvalue problems, and Lagrange’s equations. The remainder of chapter 18 utilizes the finite element method for the dynamic response of beams, trusses, and frames.

In this textbook analytical methods are developed for the response and failure of the primary structural components of aircraft. Newton's laws of motion, Hooke s law, and the first law of thermodynamics are the basis to model the thermoelastic response of thin-walled, straight bars and coplanar curved bars. Analytical methods include energy principles to develop Castigliano s theorems and to develop the cross-sectional material law for transverse shear and torsion. Stiffened shells typical of aircraft structures are analyzed with the thin-walled bar theory. Externally prescribed loads are due to accelerated flight and the thermal environment. Velocity-load factor (V-n) diagrams for maneuvers and gusts are described to evaluate flight loads. Initiation of failure is predicted by one of the following criteria: von Mises yield criterion for ductile metals; the critical load to cause buckling (failure by excessive displacements); fracture criteria for the critical stress to cause crack propagation; Puck’s criterion for the brittle failure modes in fiber-reinforced polymer composites (FRP).

The subject of structural stability of discrete conservative systems introduces the methods of stability analysis, classification of bifurcation buckling problems, the concept of imperfection sensitivity, and snapthrough at a limit point. Static instability of an elastic column from pre-buckling equilibrium, buckling, and through initial post-buckling is presented in detail. Buckling of flat rectangular plates subject to compression and shear is presented in a qualitative way using the classic charts from the National Advisory Committee for Aeronautics (NACA). The analysis for the static instability of a wing in steady incompressible flow, or divergence, is part of the discussion of aeroelastic phenomena.

• Results from linear elastic fracture mechanics (LEFM) are introduced to illustrate the relation between crack size and the stress to cause crack propagation. Airplane damage-tolerant design is based on LEFM such that subcritical length cracks do not grow to critical length between inspection intervals.
• The incentive to study optimal design is illustrated by the example of an aluminum wing spar. The objective is to achieve minimum weight by a search for two design variables. Constraints on yielding, buckling, and fracture are evaluated with the thin-walled bar theory.
• The analyses are developed for closed and open section bars made from fiber-reinforced polymer composites. The cross-sectional compliance matrix for bars with a closed cross-sectional contour and an open cross sectional contour include shear-extension coupling. The first ply failure envelope for a graphite epoxy circular tube subject to an axial force and torque is determined by Puck’s intralaminar criterion. Interlaminar failure, or delamination, is modeled with fracture mechanics, and the method is illustrated by analyses of standard fracture test specimens.
• Numerical methods for static analysis begin with the direct stiffness method, which originated to model skeletal structures consisting of bars connected by joints. Applications include coplanar trusses, beams and coplanar frames. The finite element method is developed from the integral formulation of the ordinary differential equations of an axial bar and a beam.
• Analyses for the linear elastic, dynamic response of axial bars, coplanar trusses, beams, and coplanar frames are presented using the finite element method and the mode-separation method. Hamilton’s principle and Lagrange’s equations are developed for discrete mechanical systems.
• Numerous examples to illustrate the application of the structural analysis are presented in each chapter using either U.S. customary units. or SI units.

Suggested citation
Johnson, Eric R. (2022) Aerospace Structures. Blacksburg, VA: Kevin T. Crofton Department of Aerospace and Ocean Engineering. https://doi.org/10.21061/AerospaceStructures. Licensed with CC BY NC-SA 4.0. https://creativecommons.org/licenses/by-nc-sa/4.0

Acknowledgments
The peer-reviewed work is made possible in part by financial and in-kind contributions from the Open Education Initiative at Virginia Tech, Virginia Tech Publishing, and VIVA—The Virtual Library of Virginia.

Contributors
Co-investigators: Mayuresh Patil, Rakesh Kapania
Managing editor and co-investigator: Anita Walz
Alt text writer: Joseph Brooks
Alt text assistant: Claire Colvin
Cover design and selected graphics: Kindred Grey

About the author
Eric Raymond Johnson is emeritus professor of aerospace and ocean engineering at Virginia Tech. He earned his doctoral degree in applied mechanics from the University of Michigan in 1976, and from 1976 to 2003 was a member of the engineering faculty at Virginia Tech. Dr. Johnson's research area is composite structures. Research activities include the mechanics of the response and failure of advanced composite material structures with applications to flight and land vehicles, buckling and post-buckling of plates and shells, progressive failure analysis for the prediction of energy absorption in laminated composites and in bonded joints, and fracture mechanics. He has sixty-four publications in structural mechanics, and has been awarded research funding from government agencies and industries.. He is a senior member of the American Institute of Aeronautics and Astronautics and a member of the American Society of Mechanical Engineers.