User:Eas4200c.f08.ZYX/HW Week 1-2

 Lecture Summaries:  EAS 4200c - Group ZYX

Course Description
This course, EAS4200C Aerospace Structures, will help students to achieve a better comprehension of mechanics and advance their problem-formulating skills, in response to a recent movement away from ad hoc techniques and towards the finite element method. Topics covered in this course include stress, strain, torsion, bending, shear and failure. There will also be an introduction to composite materials, an important new category of materials in aircraft construction.

Textbook
C.T. Sun, Mechanics of Aircraft Structures Wiley, 2nd edition (April 28, 2006), ISBN-10: 0471699667.

Week of 2008-08-25
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Goal
The goal of aerospace design is to build aircraft that are light and strong so we try to use materials with:




 * High Stiffness: Where stiffness is defined by Young's Modulus, $$\displaystyle E$$, which is defined by a linear relationship between stress $$\displaystyle \sigma$$ and strain $$\displaystyle \epsilon$$ :
 * $$ E = \frac \sigma \epsilon $$


 * High Strength: Where strength is defined by yield stress and ultimate stress (See Fig.1)
 * Yield Stress = $$\displaystyle \sigma_Y$$
 * Ultimate Stress = $$\displaystyle \sigma_u$$

(Yield stress is the stress at the point of transition from Elastic to Plastic deformation). Stress in Plastic Response: Increasing = Hardening ; Decreasing = Softening


 * Low Density: We simply attempt to use less dense materials than more dense ones. Density $$\displaystyle \rho$$ is mass over volume:
 * $$\displaystyle \rho = \frac m V$$


 * High Toughness: Where toughness is the ability to resist fracture and damage, and can be found by integrating the stress-strain curve using the following formula:
 * $$ \frac \mbox{energy} \mbox{volume} = \int_{0}^{\epsilon_f} \sigma\, d\epsilon $$


 * High Durability: Where durability is the ability to withstand fatigue.
 * Resistance to Corrosion: We do not want our aircraft to break down easily due to the environment.

Examples
Examples of Materials with:
 * High stiffness and high strength: Steel alloys
 * High stiffness and low toughness: Glass
 * Low stiffness and high toughness: Nylon, Plastics
 * High stiffness and high toughness: Composite Materials



Most aircraft structures use a Monocoque or Semi-Monocoque structure. However, design is also limited by aerodynamic considerations, i.e., drag and lift.

Aircraft are made of many different materials for various purposes. For example, in the F/A-18 Hornet:


 * Aluminum is used for the skin
 * Steel is used for the landing gears
 * Titanium is used for engine encasing and structural portions
 * Carbon/Epoxy is used for the fuselage and aileron.

The Boeing 787 is made of 50% composite material in order to save weight.

Problem 1.1 Set Up
The problem we are faced with is determining the optimal ratio between side lengths of a thin walled shell beam of rectangular shape so that the load bearing capacity is maximized. The walls are of a uniformed thickness, width t, which is very small when compared to the width and height of the rectangular beam itself, a and b respectively. Assumptions are also made that the magnitude of the torsional forces and the bending moment are equal: $$\displaystyle T = M $$. Furthermore the magnitude of the allowable shear is twice that of allowable stress: $$\displaystyle \sigma_{allowable} = 2\tau_{allowable} $$

The Shear Stress at any given point in a wall of the beam is governed by the equation:


 * $$ \tau = {V \over t}$$

where:


 * V = shear force resultant
 * t = thickness of beam wall at that point

Because of the thin walls we can safely assume that shear forces are being distributed across all walls uniformly, also known as Shear Flow. The shear stress found in the beam is represented by the following formula:


 * $$ \tau = {T \over 2abt}$$

where:


 * T = torsional force in the beam
 * a = width of the beam
 * b = height of the beam
 * t = thickness of beam walls

To determine T for the total beam we can simply sum the separate T forces found in the individual walls:


 * $$\displaystyle T = T_{ab} + T_{bc} + T_{cd} + T_{da} $$

We can easily determine that $$\displaystyle T_{ab} = \frac {\tau a b t} 2 $$ and since the same rule can be applied to all the other segmented T's the total T is found.

There are then two routes that must be undertaken to determine the proper ratio of b/a for maximal loading capacity. The first of these is to assume that the $$\displaystyle \sigma_a $$ is the first allowable stress point that is reached. Using that as a boundary condition we will then have to verify that the resultant $$\displaystyle \tau $$ is less than $$\displaystyle  \tau_a $$

Contributing Team Members
Robert Reger Eas4200c.f08.ZYX.reger/Week1,2

Casey Barnard

Wei-Teck Lee Eas4200c.f08.zyx.lee 02:57, 19 September 2008 (UTC)

Andrew Fahrney Eas4200c.f08.ZYX.fahrney 18:51, 19 September 2008 (UTC)

Brian O'Mahoney Eas4200c.f08.ZYX.O&#39;Mahoney 22:38, 19 September 2008 (UTC)