User:Eas4200c.f08.aeris.krammer/HW1.2

Purpose and the Big Picture (Aerospace Structures EAS4200C)  The purpose of the class methodology of teaching is to allow for easy access to vast amounts of information while in the process, learn the knowledge that is being archived and presented. By using Wikiversity as a means of infomation exchange, not only can the class benefit from each other by learning from different perspectives in a synergetic approach, the entire engineering population can benefit as well. This approach is similar to the MIT open course ware. It basically provides lecture notes to the general public in a free and organized manner so that information can be shared to the masses. Throughout the semester, the class and the group will learn and develop communication skills, presentation skills, and essential reporting-writing skills. The technical goals of this course includes understanding the structures and components of an aircraft, understanding the mechanics of these components and the derivation of the formulas in which the mechanics can be described, understanding how and why components fail, fracture, and buckle, etc.

Preface and Introduction to Aerospace Structures  The book "Mechanics of Aircraft Structures" by C.T. Sun aims to teach students to better understand the mechanics of aircraft structures, formulating the problem, and judging the correctness of the solution. It also exposes the student to an important analysis topic used in engineering for the last three decades: fracture mechanics. This is important because fracture mechanics has been used to study the durability and damage tolerance on aircrafts. The book also introduces the topic of composite materials. Since recently advanced materials have been used in building aircrafts, it is imperative for the student to realize the difference between these new materials from the traditional metallic materials. Since the finite element method is becoming more and more prevalent in the analysis of structures, the book begins to emphasize multi-dimensional stress, strains and their relationships. Also, displacement instead of stress and strain will be used to derive the governing equations of these structures. This teaching methodology will be taught along with the classical plane stress and plane strain concepts taught in mechanics of materials.

Aerospace structures include everything from wings to fuselage, from stabilizers to engine casings. Modern aerospace structures are made from various types of materials ranging from metal alloys to composites. Aerospace structures differ from other structures due to the highest demands for performance and light weight. The structure's geometry and material dictate the stresses it can withstand during flight. Analysis of aerospace space structures provides information on areas of critical stress and strain. Analysis methods include the finite element method coupled with knowledge of material failure criteria.

Material In aerospace science, the design of the system must be light-weight and strong. It must be light-weight to ultimately improve efficiency. The lighter the aircraft, the less fuel it uses or the more payload it can carry. It must be strong to withstand the rigorous forces and stresses induced on the system during flight. Thus, the materials chosen to build these structures must have high stiffness, high strength, and a high strength to weight ratios. To understand the rationale for using materials with high stiffness, strength, etc., stiffness, strength and strength to weight ratios must be defined. The stiffness of a material is essentially how sensitively the material elastically deforms when stress is induced on it. It is known as Young's Modulus or the Modulus of Elasticity. It is the slope of the stress versus strain line where $$\displaystyle E$$=$$\displaystyle \sigma$$$$\displaystyle \epsilon$$. In this equation, $$\displaystyle E$$ represents Young's Modulus, $$\displaystyle \sigma$$ represents stress (which is force per unit area) and $$\displaystyle \epsilon$$ represents strain (which is the change in length over the original length). Strength is defined as the stress at which the material yields. Yielding is where plastic deformation occurs and the material can no longer return to its original shape. The strength to weight ratio is the ratio of the yield stress to the materials density. The toughness of the material is also an important characteristic. Aircraft structures want to have high toughness as well. Toughness is defined as the ability to resist fracture. High stiffness and high strength materials include steel, titanium, and aluminum alloys. An example of high stiffness and low toughness is glass. Once glass cracks, the crack can easily propagate through the material and thus it has low toughness. Low stiffness and high toughness materials include plastics, nylon, and aluminum as aircraft skin. High stiffness and high toughness materials include fiber reinforced composites like epoxy. Materials are used tot he full capacity meaning that the stresses subject to the material should be close to the yield stress in order to the the aircraft to be more weight efficient.

Geometry The design of an aerospace structure is not only subject to the material but also to the geometry of the structure itself. Obviously, geometrical configurations that exhibit strength are required. Structures called monocoques and semi-monocoques are examples of strong geometrical configurations. The monocoque is a "construction technique that supports structural load using an object's external skin" (source:[]). The semi-monocoque is a technique where a something like a structural beam is inserted into a monocoque to prevent buckling. The design is also limited by geometry. For instance, the airfoil of an aircraft must be "chosen according to aerodynamic life and drag characteristics" (source:"Mechanics of Aircraft Structures", C.T. Sun).

Aircraft Example "Because of their high stiffness/weight and strength/weight ratios, aluminum and titanium alloys have been the dominant aircraft structural materials for many decades. However, the recent advent of advanced fiber-reinforced composites has changed the outlook" (source:"Mechanics of Aircraft Structures", C.T. Sun.) The F18 Hornet fighter jet is a good example of how and where materials are used on an aircraft. It has aluminum skin since aluminum is fracture tough, steel landing gears, titanium engine casing, stabilator, landing and cockpit gear, and carbon epoxy fuselage, aileron (control surface along the trailing edge of the wing), stabilator and vertical stabilizer.

Problem 1.1
(Mechanics of Aircraft Structures, C. T. Sun, 2006) Problem Statement: A rectangular thin-walled beam section carrying a bending moment M and torque T is given. The total contour length is of the beam section is a given fixed value L=2(a+b). Furthermore, M=T and σallowable=2τallowable. With this information, we need to find the optimum b/a ratio for the maximum load bearing capacity and so for the most efficient section. 

To find the optimum b/a ratio for the maximum load bearing capacity, the bending moment must be maximized. The equation for a maximized bending moment is as follows: Mmax = (2σall)(I/b)max. I, moment of inertia, is calculated using the length of the base, b, and the height, a. Using the given formula above, L=2(a+b), a can be written in terms of b,; therefore, I can be written in terms of b alone and plugged into the bending moment equation, Mmax = (2σall)(I/b)max. Finding where the change of the bending moment with respect to b equals zero gives the value of b where the bending moment is the highest. For example: $$dM/db = 0$$.

Using the optimum b value, the optimum a value can be found through the first assumption, L=2(a+b). The optimum b value is divided by the optimum a value to obtain the optimum b/a ratio.

Assumptions:
 * $$\ t << a $$ (given)
 * $$\ t << b $$ (given)
 * $$\ L = 2 (a+b) $$ (given)
 * $$\ M = T $$ (given)
 * $$\ \sigma_{allowable} = 2 \tau_{allowable} $$ (given)

Governing Equations:
 * $$\ \sigma = \frac{T}{2 a b t}\,$$

This problem must be solved for 2 separate cases to arrive at a conclusion. The first case is where $$ \sigma_{max} = \sigma_{allowable}$$, and the second case is where $$\tau_{max} = \tau_{allowable}$$.

Case 1: Assume $$\sigma_{max} = \sigma_{allowable}$$.

Recall that $$\sigma = \frac{M z}{I}\,$$ with $$z = \frac{b}{a}\,$$

We can now solve for M, which as can be seen below is a function of the ratio of I to b.
 * $$M = \frac{2 I \sigma_{max}}{b}\,$$
 * $$M = \frac{2 I \sigma_{allowable}}{b}\,$$
 * $$M = 2 \sigma_{allowable}* \frac{I}{b}\,$$

In order to maximize M we need to either maximize I, minimize b, or do a combination of both. Before we look at this further, we need to get an expression for I in terms of the given variables.


 * $$ I = \sum_{i=1}^4 \frac{b_i h_i^3}{12}\ + A_i d_i^2$$
 * $$ I = 2[\frac{a t^3}{12} + a t (\frac{b}{2})^2] + 2(\frac{t b^2}{12})$$
 * $$ I = \frac{a t}{12} (t^2 + 3 b^2) + 2(\frac{t b^2}{12})$$

Going back to the given assumptions, we recall that
 * $$\ t << b $$ --> $$\ t^2 << b^2 $$ --> $$\ t^2 << 3 b^2 $$

We can now assume that the wall thickness is negligible, and neglecting the t term we can now substitute our I back into the term we are trying to maximize, $$ \frac{I}{b}$$.


 * $$ \frac{I}{b} = \frac{t b}{6} (3 a + b)$$

Recalling that $$ a = \frac{L}{2} - b)$$, we can simplify to just one variable, b.


 * $$ \frac{I}{b} = \frac{t b (3 L - 4 b)}{12}$$

We note that this is a downward opening parabola, and using some simple calculus we solve for the vertex at $$ b = \frac{3 L}{8}$$.

Substituting back into the original assumption $$\ L = 2 (a+b) $$, we find that $$ a = \frac{L}{8}$$.