User:Eas4200c.f08.vqcrew.c/Homework 1

Administrative Considerations for EAS4200C
The general structure of the course is formulated to cultivate a collaborative setting in academia for students to prepare for working in similar situations in industry. This is achieved through assigning primarily group-based tasks, chiefly in the form of regular homework reports created and submitted using a MediaWiki-based platform (in this case, Wikiversity). The intent is that groups of 5 or 6 students will collaborate on a regular basis to create the reports, which contain summaries of important lecture topics as well as traditional homework problems and solutions. These reports are advantageous not only through their fostering of teamwork and collaborative abilities but also by encouraging proper citation of references and sources, which is a key skill in industry. As an incentive to fully exploit the value of this resource, the reports constitute 31% of the final grade in the course, the remainder of which is split evenly amongst 3 exams.

Aside from collaboration, the emphasis in the course is to develop both an understanding of mechanics and good problem-solving skills. A key to this development is an ability to formulate problems and subsequently judge the correctness of a solution to these problems. Approaching the course in this fashion avoids the old “ad-hoc” methods of structural analysis. Additional information about the aims of the course and its structure can be found at the course website.

Overview of Using Wikipedia Responsibly
During lecture, Dr. Vu-Quoc pointed out several of the advantages of using Wikipedia, how to use it responsibly and how to avoid vandalism. Various administrative issues were discussed such as how to gain access to the Wikiversity page, the format for usernames and where and how to post information on the group's user namespace. Examples were given of proper and improper ways to post and edit material and suggestions were offered such as using the hide feature for derivations of equations, when applicable. Dr. Vu-Quoc went over pages that he had created to provide an idea of what was expected. He also showcased examples of vandalism to stress the importance of not editing unrelated articles so as to avoid drawing attention to ourselves. He also illustrated the process of how to retrieve a vandalized page in the case, which would be helpful if one's page should be vandalized to restore it to the original form.

He additionally explained the importance of not creating actual Wikipedia articles for the class assignments, but rather to edit only the user namespace. The process of homework would be done this way to avoid Wikipedia's review process when a Wikipedia page is created. He went over the usefulness of drop-boxes and displayed the simplicity of copying and pasting from one's user namespace to the group's namespace. Additionally, he explained the importance of turning in link that came from the history rather than the actual page to avoid any confusion. This would remedy the problem of someone changing something in the article after the homework has been submitted.

Preface of the Text
In the preface of the first edition of Mechanics of Aircraft Structures written by C. T. Sun, the author first explains that the text is written for third and fourth year aeronautical engineering students. The author assumes that the reader has already taken the mechanics of materials course in their curriculum. The text is written for a one semester course.

The author then proceeds to outline that the book will deviate from the traditional structures text's "ad hoc techniques" of problem solving. The industry has shifted to using finite elemental methods for structural analysis. The text explains that it will emphasize the understanding of structures so that numerical data given by modern finite elemental methods will be interpreted correctly. Another difference is that the author derives the governing equation via displacement rather than stress and strain.

Secondly, fracture mechanics are discussed in the text, giving the student exposure to this topic prior to graduate school. The author states that fracture mechanics have been a useful tool in the past thirty years of aircraft structures. The topic of fracture mechanics is explained by "Griffith's concept of strain energy release rate." The author feels that calculating the strain energy release rate won't be a difficult problem for the third and fourth year students to calculate. Calculating this also reveals the change of the stored strain energy before and after the fracture.

Lastly, with the recent development of composite structures in industry, it gives an introduction to the properties of composite materials. Until now, traditional metals have been studied in the curriculum, and relative to the metals previously studied, composites are significantly different. The text covers the material properties of composites such as the stress-strain curves of anisotropic and further covers the analysis of symmetric laminates of composite materials. The author emphasizes that this is merely an introduction to the subject on composite materials, and isn't a substitute for a composites course.

The preface also mentions that Chapter 7 will also include buckling concepts at an elementary level. The author states that postbuckling strengths of bars and panels are commonly used in aircraft structures and this chapter exposes the reader to the concept without the necessity of more advanced courses.

In the second edition the author corrected typographical errors and add to the materials in chapters 3, 4 and 6. Chapters 1 and 2 were not modified and in the latter chapters he covers new topics including an introduction to the effect of plasticity on fractures, Timoshenko beam theory and other principles as well.

Introduction to Aerospace Structures
The governing equations in aerospace structures and structural composition in general are governed by partial differential equations. These equations can be solved by the finite elemental method (FEM), which is used throughout the engineering discipline as well as structural composition.

The emphasis in this course is to develop problem solving skills while understanding mechanics. The finite elemental method will numerically solve the partial differential equations so it will be the job of the future engineer to interpret the numerical results to judge the correctness of the results. In addition, the ability to formulate problems and avoid the more traditional method of structural analysis will be emphasized throughout the course.

The main consideration during the aircraft design process is to reduce weight while maintaining strength in the structure. Strength, stiffness, and toughness all play important roles in the overall performance of an aircraft. The role of a design engineer is to create a balance of these properties that best satisfies all of the requirements of the aircraft.




 * Strength - the resistance of a material to deformation and rupture.
 * The yield stress, $$\displaystyle \sigma_Y$$, of a material is the point at which the material will begin to plastically and permanently deform.
 * The ultimate stress, $$\displaystyle \sigma_u$$, of a material is the critical point at which the material fails catastrophically.
 * Stiffness - the ability of a material to resist deformation caused by forces acting on the object.
 * The strain, $$\displaystyle \epsilon$$, of a material is the measure of the deformation caused by an applied force.
 * Young's modulus, $$\displaystyle E$$, is the measure of a material's stiffness. Young's modulus is the ratio of the stress applied over the strain caused in response, $$\displaystyle E = \sigma/\epsilon$$.  Figure 1 displays the stress-strain curve for steel.  Young's modulus represents the slope of the curve in the linear, or "elastic", region.  In this region, no permanent or "plastic" deformation has taken place and the material will return to its original dimensions.
 * Toughness - the overall resilience of a material and resistance to fracture. Toughness is measure in units of energy per unit volume a material can withstand before failure.

Examples of Materials and Their Properties
 Glass  - high stiffness, low toughness. Unlikely to bend or deform, yet vulnerable to fracture.

 Plastics  - low stiffness, high toughness. Easily deformed, but resistant to rupture.



 Metals  - Aluminum has high toughness, but is weak in terms of stiffness. Aluminum will not resistant deformation under high loads. Steel alloys are better suited for a situation requiring high stress handling capabilities.

All of these materials have their strengths and weaknesses. All of these materials can be used safely and effectively as long as these constraints are recognized. In the search for highly stiff, highly tough materials, the strengths of different materials were combined and their weaknesses diminished. Materials known as composites are formed from a network, or "matrix", of strong, flexible fibers held in place by stiff epoxy. The union of these materials creates a new, strong and lightweight material ideal for the aerospace industry.

In order to optimize performance, aerodynamic considerations must be made during the design process. Even though a material is structurally robust, there many be other factors that make that material detrimental to the overall function of the aircraft. For example, a material with vulnerability to cold or corrosion could not be used on the outer skin of an aircraft. The shape of major features of the aircraft must function both structurally and aerodynamically. The fuselage must be strong enough to withstand years of stress associated with pressurizing and depressurizing the aircraft while also maintaining a small cross-sectional area to keep aerodynamic drag to a minimum. The wings must be shaped to create lift to allow the aircraft to fly, but they must also act to support the weight of the fuselage.

Further considerations can be found at the online notes.

Problem 1.1
Given Figure 2, find the optimum ratio $$\frac{b}{a}$$  to maximize the load bearing capability of the thin-walled beam.

The assumptions for this problem are as follows:
 * 1) The cross-sectional perimeter is constant, i.e. $$L=2(a+b)=Constant$$
 * 2) The torque $$T$$ is equal to the bending moment $$M$$, i.e. $$T = M$$
 * 3) The maximum allowable bending normal stress is twice the maximum allowable shear stress, i.e. $$\sigma_{allow} = 2\tau_{allow}$$

The assumption that $$t$$ is very small compared to $$a$$ and $$b$$ is also valid.

Since the cross-section walls are very thin, the shear stress can be assumed to be uniform along the walls, as shown in the shear flow diagram in Figure 3. The shear distribution and resultant shear vector $$V$$ for a small piece of the wall is also shown. This distribution is also mechanically equivalent to the linear distribution shown in Figure 4, where $$\tau =\frac{V}{t}$$.



The total torque ($$T$$) can be described as the sum of the torques on each wall of the box beam as follows: $$T = T_{AB}+T_{BC}+T_{CD}+T_{DA}$$

$$T_{AB}=(\frac{b}{2})Va=\frac{1}{2}\tau tab$$

$$T_{BC}=(\frac{a}{2})Vb=\frac{1}{2}\tau tab$$

$$T_{CD}=(\frac{b}{2})Va=\frac{1}{2}\tau tab$$

$$T_{DA}=(\frac{a}{2})Vb=\frac{1}{2}\tau tab$$

Therefore, $$T=2t\tau ab$$, and $$\tau =\frac{T}{2abt}$$.

There are two cases that must be analyzed in order to solve the problem:
 * 1) Assume the bending normal stress (σ) reaches $$\sigma_{allow}$$ first, then verify that $$\tau \le \tau_{allow}$$.
 * 2) Assume the shear stress (τ) reaches $$\tau_{allow}$$ first, then verify that $$\sigma \le \sigma_{allow}$$.

Case 1

Case 1 begins by recalling the formula for bending normal stress: $$\sigma =\frac{Mz}{I}$$, where $$M$$ is the bending moment, $$z$$ is the ordinate of a point on the axis perpendicular to the bending neutral axis, and I is the second area moment of inertia defined by $$I=\iint_A z^2 \ dy\, dz$$. Figure 5 shows each item pictorially.



HW1 Contributing Team Members
The following students contributed to this report: John Saxon Eas4200c.f08.vqcrew.c 03:26, 19 September 2008 (UTC) Adam Edstrand Eas4200c.f08.vqcrew.a 03:43, 19 September 2008 (UTC) Javier Stober Eas4200c.f08.vqcrew.b 13:22, 19 September 2008 (UTC) Darin Toscano Eas4200c.f08.vqcrew.d 04:02, 19 September 2008 (UTC) Kevin Klauk Eas4200c.f08.vqcrew.E 12:21, 19 September 2008 (UTC)