User:Eas4200c.f08.aero.e/Week 2

=Wednesday=

Material Properties

Weight is a critical component in the design of aircraft structures. The weight in the fuselage, wings, and engines is the primary concern. Current design is a balance between performance and how much it will cost to lift off the ground. Ideally materials used in aerospace applications have high stiffness and strength while still remaining light weight.

Stiffness is described by Young's Modulus ($$ \displaystyle E = \displaystyle \epsilon/ \displaystyle\sigma $$) the ratio of stress over strain. A higher Young's modulus corresponds with a higher material stiffness.

NOTE: Young's modulus is valid only in the elastic range (shown below).

The strength of a material is measured by the materials yield stress ($$\displaystyle \sigma$$y). Obviously a higher yield stress corresponds with a stronger material.

A materials toughness represents its ability to resist fracture. This is also know as fracture toughness.

Shown above is a graph of stress versus strain for a typical metal. Note that the graph is not exact and only for conceptual purposes. From the graph it is apparent that the ratio of stress and strain is linear inside the elastic range. Then plastic deformation and permanent deformation. If the material is stressed beyond its maximum yield point it will reach rupture stress and fail. Inside the plastic range a positive slope represents hardening of the material while a negative slope represents softening.

The Modulus of elasticity also known as young’s modulus is the rate of change between strain and stress. Further informations can be found at

http://www.instron.us/wa/resourcecenter/glossaryterm.aspx?ID=99&ref=http://www.google.com/search.

The chart below lists various categories of material properties and the materials which possess those properties.

The structure of an aircraft has two primary components:


 * 1) Geometry - Common aircraft geometries include monocoque shell like constructions and semimonocoque stiffened shell like structures.
 * 2) Material - Common aircraft materials include aluminum, titanium, and composite materials.

Below is a picture of the Boeing 787 Dreamliner which illustrates the aircraft's material composition. Composite materials make up fifty percent of the 787's total weight.

Boeing 787

=Friday=

Problem 1.1

Problem 1.1in "Mechanics of Aircraft Structures" 2nd Ed. by C. T. Sun states:

The beam of a rectangular thin-walled section (ie., t is very small) is designed to carry both bending moment M and torque T. If the total wall contour length L = 2(a + b) is fixed, find the optimum b/a ratio to achieve the most efficient section if M = T and σallowable = 2τallowable. Note that for the closed thin-walled sections such as the one in Fig. 1.16, the shear stress due to torsion is $$ \tau = {T \over 2abt} $$



The problem asks to find the ratio of the height vs. the width of a shell beam. To find this, first some assumptions must be made.


 * 1) M = T
 * 2) σallowable = 2τallowable
 * 3) L = 2(a + b)
 * 4) We can also assume shear stress distribution uniform across the wall, because the cross-section walls are very thin.

 Methodology  The shear stress can be represented with the figure below.



$$ T = T_{AB} + T_{BC} + T_{CD} + T_{DA}$$

The force can be shown as a parabola, like water in a pipe, like Fig. A below. This will translate to the next force below to B and C.



$$ T_{AB} = {b \over 2} vb = {1 \over 2} \tau abt $$

 Case 1: 

For case one, we assume σ (bending normal stress) reaches σallowable and then verify that τ less than or equal to τallowable

Recall:       $$ \sigma\  = {M_z \over I} $$

Where M is the bending moment, z is the of a point on axis parallel to the normal bending axis, and I is the 2nd moment of inertia. The equation for the moment of inertia is:

$$ I = \iint_{A} \,z^2 dy\, dz = {bh^3 \over 12}$$

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