User:Eas4200c.f08.spars.rondon/Homework 1

[[Image:Lockheed_SR-71_Blackbird.jpg|thumb|right|]

=Professor's Notes= The forces governing aerospace structures can be described by partial differential equations which can be solved through implementation of finite elemental method (FEM). It is therefore the goals of EAS 4200C to; develop problem solving skills stemming from an understanding of aerospace mechanics, instill the ability to formulate aerospace mechanics problems, and judge the correctness of the arrived at solution all while avoiding the older ad hoc techniques of structural analysis. This class also hopes to implement cooperative learning in a manner that reduces the stress on the individual while also promoting the communication skills essential to work in the aerospace field.

=Class Notes= The vision for the class is broken up over several topics. These include the technique of solving partial differential equations known as the finite element method-which is used in every field of engineering to date-as well as aerospace structures, which is governed by partial differential equations. The emphasis of this course will comprise of developing both the skills needed to become effective engineers and the methodologies that can be used to understand, formulate, and judge the solutions to the problems we will be working on. Students are broken up into 5-6 member teams, that collaboratively work on and submit homework assignments. Students will be using the features of mediawiki, textbooks from wikiversity, information from MIT's Open Courseware and wikipedia, all of which are a part of the big picture that makes up the course to accomplish this. The goal of the configuration of the course is to be a collaboration between the team of students, the whole of the class, as well as the professor. Each team member has a username that can be used to identify them by the instructor, and allows the instructor to keep track of each member's contribution to the overall effort. By increasing the incentive to learn collaboratively, the instructor hopes to have an overall positive impact on improving the learning method in which each student participates.

=Text Book= Mechanics of Aircraft Structures, written by C.T. Sun, is intended for Aeronautical Engineering students with existing knowledge in courses such as Mechanics of Materials.

Preface
The second edition preface simply names the purpose of creating a second edition. The second edition's purpose is to "correct a number of typographical errors", "add more examples and problems for the student, and introduce a few new topics." The author then thanks those who contributed to the changes. The preface then includes the first edition's preface, which gives the overall introduction to the book. The author explains that because of the development of finite element codes, mathematical techniques which were highly emphasized in the engineering curriculum in the past, are now obsolete. Because of this, the book's material is presented with a greater emphasis on the understanding of the theoretical concepts, how to structure problems, and then judge their solutions. This also means that the text places a minor emphasis on the mathematical steps for finding the solutions to the problems.

The text also includes introductory information for two recent, important developments in the industry. The first is fracture mechanics, which the book describes as "the most important tool in the study of aircraft structure damage tolerance and durability in the past thirty years." (p. XV) It is explained that this material is vital for us to know upon graduation since it is an industry standard, having previously been knowledge reserved specifically for graduate school. As such, the text is attempting to correct this. The second topic the book also incorporates, in addition to the information found in rival texts that cover this subject, information on composite materials.

1.1: Introduction
"The main difference(s) between aircraft structure materials and civil engineering structure materials lies in their weight." (p. 1) Weight is the number one consideration in the design of an aircraft. Because one can't afford to fly around with excess weight, not only does the aircraft have to be designed out of materials with high structural property values with low weights, but the individual components have to be pushed to their limits in order to ensure every pound of material is used to its fullest capability.

“Aircraft structures must be designed to ensure that every part of the material is used to its full capability. This requirement leads to the use of shell-like structures (monocoque                     constructions) and stiffened shell structures (semimonocoque constructions).” (Sun, Page 1)

Along with material characteristics, nonstructural considerations (i.e. aircraft size, shape, lift, drag) have to be considered. Since there are so many factors to consider, special materials are developed for these special applications. The dominant aircraft structural materials have been aluminum and titanium due to their high stiffness/weight and strength/weight ratios. However, in recent years, engineers have realized that they can obtain 30-40% weight savings by switching to composites such as carbon fiber, and have slowly been doing so. According to the text (p. 2) composites account for 50% of the weight of the Boeing 787.

Additionally, on top of pushing the components' materials to the limit, the design is further complicated because often, a component is designed with application in mind, rather than structural consideration. For example, an airfoil is designed specifically to provide lift, not necessarily to be extremely rigid. This means that the solutions to problems encountered during the design process are extremely limited, and because of this we tend to use exotic building materials. (p. 1)

1.2: Basic Structural Elements in Aircraft Structure
“Major components of aircraft structures are assemblages of a number of basic structural elements, each of which is designed to take a specific type of load, such as axial, bending, or torsional load.”(Sun, Page 2)

This section covers the basic equations and background information for understanding the purpose with which certain parts of the airplane are designed. Every component has a very specific application, and as such, has certain characteristics whose abilities have to be maximized by the part's design.

1.2.1 Axial Member
Axial members are used to carry extensional or compressive loads applied in the direction of the axial direction of the member. The resulting stress is uniaxial:


 * $$\ \sigma = \mathbf{E}\varepsilon \,$$

where E and ε are Young's modulus and normal strain, respectively, in the loading direction. The total axial force F provided by the member is


 * $$\ \mathbf{F} = \mathbf{A}\sigma = \mathbf{E}\mathbf{A}\varepsilon \, $$

where A is the cross-sectional area of the member. (p. 2)

"EA" is known as the axial stiffness of the component, and as one can see, its value is determined by both the modulus of the material the part is made of, and the cross-sectional area at a given point. It should be noted that as long as two different components have the same material, and the same cross-sectional area (regardless of shape) then they have the same axial stiffness.

Axial members tend to be slender, and because of this, they're particularly vulnerable to buckling when under compression. "Buckling strength can be increased by increasing bending stiffness and by shortening the length of the buckle mode." (p. 2) Also, proper selection of the axial member may also help in reducing buckling. For example, a channel section has a higher bending stiffness than a circular one, so it tends to prevent buckling more effectively. However, in practical applications, most axial members, such as stringers are so slender that no design can provide a buckling strength great enough to avoid buckling. In order to remedy this, lateral supports are placed along the member to provide more strength. Some of these are ribs in wings, and frames in the fuselage. (p. 3-4)

1.2.2 Shear Panel
A shear panel is a thin sheet of material used to carry in-plane shear load. Consider a shear panel of uniform thickness t under uniform shear stress τ ... The total shear force in the x-direction provided by the panel is given by


 * $$\ \mathbf{V_x} = \tau\mathbf{t}\mathbf{a} = \mathbf{G}\gamma\mathbf{t}\mathbf{a} \, $$

where G is the shear modulus, and γ is the shear strain. Thus, for a flat panel, the shear force Vx is proportional to its thickness and the lateral dimension a. (p. 4)

The shear force in a curved panel under stress τ can be broken up into a horizontal component, Vx, and a vertical component, Vy. These resulting forces only depend on the dimensions of the curved panel, and not the actual contour of it. If the contour doesn't affect the capacity of a member to handle shear stress, then a more efficient use of material would be to use a straight panel. (p. 5)

1.2.3 Bending Member (Beam)
A structural member that can carry bending moments is called a beam. A beam can also act as an axial member carrying longitudinal tension and compression. According to simple beam theory, bending moment M is related to beam deflection w as


 * $$\ \mathbf{M} = \mathbf{-E}\mathbf{I}\frac{d^2\omega}{dx^2} \, $$

where EI is the bending stiffness of the beam. The area moment of inertia I depends on the geometry of the cross-section. (p. 5)

Since transverse shear forces tend to produce bending moments, beams tend to be designed with the capability to sustain both kinds of forces in mind. (p. 5) Additionally, for beams with large span/depth ratios, bending stress usually tends to be a more critical design criterion than transverse shear stress. (p. 5-6)

For a beam supported by a wall at one end, with a shear force acting on the opposite end, of cross-sectional height h and width b, we can define the maximum normal stress as:


 * $$\ \sigma_\max = \frac{\mathbf{M_\max (h/2)}}{\mathbf{I}} = \frac{\mathbf{V}\mathbf{L}\mathbf{(h/2)}}{\mathbf{bh^3/12}}= \frac{\mathbf{6VL}}{\mathbf{bh^2}} \, $$ (p. 6, eqn. 1.7)

The profile of the magnitude of the transverse shear stress distribution follows a parabola along the depth of the beam with a maximum value along the neutral plane of:


 * $$\ \tau_\max = \frac{3}{2} \frac{\mathbf{V}}{\mathbf{bh}}\, $$ (eqn. 1.8)

From the ratio


 * $$\ \frac{\sigma_\max}{\tau_\max} = \frac{\mathbf{4L}}{\mathbf{h}} \, $$ (eqn. 1.9)

one can see that for large span-to-depth ratios, such as in wings, the bending stress if far more important than transverse shear stress. For these kinds of beams then, the goal of the design is to maximize the bending stiffness by having the most appropriate cross-section possible. (p. 6)

In order to utilize the full capacity of the material in a beam, the material must be set as far apart from the neutral axis as possible. (p. 6)

1.2.4 Torsion Member
Torque, formed by shear stress acting in the plane of a member's cross section, is an important load to consider when designing aircraft components. (p. 7)

Consider a hollow cylinder subjected to a torque T ... The torque-induced shear stress τ is linearly distributed along the radial direction. The torque is related to the twist angle Θ per unit length as


 * $$\ \mathbf{T} = \mathbf{G}\mathbf{J}\theta \, $$ (eqn. 1.10)

where J is the torsional constant. For hollow cylinders, J is equal to the polar moment of inertia of the cross-section, i.e.,


 * $$\ \mathbf{J} = \mathbf{I_p} = \frac{1}{2}\pi\mathbf{(b^4-a^4)}= \frac{1}{2}\pi\mathbf{(b-a)(b+a)(b^2+a^2)} \, $$ (eqn. 1.11)

The term GJ is usually referred to as torsional stiffness.

If the wall thickness t=b-a is small compared with the inner radius, then an approximate expression of J is given by


 * $$\ \mathbf{J} = 2\mathbf{t}\pi\mathbf{r^3} \, $$ (eqn. 1.12)

where r=(a+b)/2 is the average value of the outer and inner radii. Thus, for a thin-walled cylinder, the torsional stiffness is proportional to the 3/2 power of the area (πr2) enclosed by the wall. (p. 7-8)

Since the torsional shear that a beam undergoes is proportional to the radius, the material towards the inside of the beam isn't used to its full capacity. Take for example using a solid circular beam and a hollow tube, both of the same mass, with the beam of a diameter of 2cm, and the tube of an inner radius of 5cm with a wall thickness of 0.1cm, the hollow tube undergoes 50 times more shear than the beam. This proves that thin-walled structures may be used as extremely efficient torsion members. (p. 8)

1.3 Wing and Fuselage
"The wing and fuselage are the two major airframe components of an airplane. The horizontal                              and vertical tails bear close resemblance to the wing. Hence, these two components are                                 taken for discussion to exemplify the principles of structural mechanics employed in                                  aircraft structures." (Sun Page 8)

1.3.1 Load Transfer
Load transfer is the "passing on" of high loading from flexible members to stiffer members. Stiffeners can be added to the skin of an aircraft to avoid excessive deflection in the skin. When the skin of an aircraft body encounters an external pressure, the load can be transferred to these stiffeners, which in turn may transfer the load to other more rigid members. Understanding of load transfer is important when dealing with non-ideal situations where a bending moment or air pressure is not uniform. If the load from these things are not transferred correctly from a weak member to a stronger member, fracture may occur.

1.3.2 Wing Structure
"The main function of the wing is to pick up the air and power plant loads and transmit                      them to the fuselage." (Sun, Page 10)

The wing, along with most other parts of the aircraft body, acts as a beam and as a torsion member. The individual members of the wing include: ribs, spars, flanges, wing skin, vertical stiffeners, and so on.



Wing ribs run chordwise inside the wing and carry in-plane loads. They help keep the wing's shape and also serve to transfer loads. Spars run spanwise along the wing and are designed to take bending and shear loads. It is usually shaped with a web and flange configuration, with the flange or heavy cap to take bending.

1.3.3 Fuselage
"The fuselage structure is a semimonocoque construction] consisting                 of a thin shell stiffened by longitudinal axial elements (stringers]                  and longerons) supported by many transverse frames or rings                   along its length." (Sun, Page 12)

The fuselage is the body of an aircraft. Its purpose is for position control and stabilization of surfaces in specific relationships to lifting surfaces, required for aircraft stability and maneuverability. The primary forces acting on the fuselage are wing reactions, pay loads, reactions of the landing gear, and internal pressures due to passengers on board. To combat these forces the fuselage often has a circular cross section. This ring-like shape, along with added support (stringers), increases the buckling strength of the material used and creates small, self-equilibriated loads.

1.4: Aircraft Materials
"Selection of aircraft materials depends on many considerations...Seldom is a single material                    able to deliver all desired properties in all components of the aircraft structure. A                       combination of various materials is often necessary." (Sun, Page 14)

In the last 30 years, there have been many advances to replace the traditional metallic materials used in aircraft structures today. Those advances have brought about materials which allow for less weight and more strength; two things that are key in the structural industry. Some of the major materials are steel, aluminum, and titanium alloys, along with fiber-reinforced composites.

Steel alloys are used when there is a need for a material with high yield stress and high stress because of its high density. Steel alloys are used in landing gear units and highly loaded fittings. (Sun, 14) The only downfall with using steel is that it is extremely heavy and must be plated due to its poor corrosion resistance. Aluminum alloys are used because they have a good fatigue life as well as fracture toughness and a very slow crack growth rate. They are commonly found in the fuselage and lower wing skins due to the constant cyclic tensile and compressive stresses. Titanium alloys are used because they have superior corrosion resistance along with ultimate and yield stresses much higher than the best aluminum alloys. Titanium alloys can also be used in high heat conditions (i.e. up to 1000° F). This alloy has been used sparingly because of its high machining cost but more effective methods, like near net shape forming, have been employed to help make using titanium more economic. "Materials made into fiber forms can achieve significantly better mechanical properties than their bulk counterparts." (Sun, 15) Fiber-reinforced materials can be categorized into the three categories of polymers, metals, and ceramics. These composites are much stiffer, stronger, and lighter than the major materials and are most suitable for aircraft structures. Through various application processes, they can provide multi-directional load capabilities, corrosion resistance, damage tolerance, and high fatigue life.

Team Member Contributions
The following students contributed to this report:

Eduardo Rondon Eas4200c.f08.spars.rondon 00:58, 17 September 2008 (UTC)

Eas4200c.f08.spars.prey 03:12, 18 September 2008 (UTC)

Michael Lee Eas4200c.f08.spars.lee

Thomas McGilvray Eas4200C.f08.spars.mcgilvray 03:31, 19 September 2008 (UTC)

=References= 1. Sun, C.T. (2006). Mechanics of Aircraft Structures John Wiley & Sons, New York. ISBN 0471699667.

2. Vu-Quoc, Loc (2008) Lecture Notes University of Florida