User:Eas4200c.f08.aero6.ortega/hw1

EAS4200c Learning Method
Homework reports will be collaborative efforts of teams of 6. Each team will use wiki software to present class activities including lectures, homework problems and additional contributions. This collaborative style allows students to assist and learn from each other while discussing the course material. The method also encourages a larger class collaboration as the best of each set of homework reports will be selected for the entire class to review. Each student will be evaluated by all of his or her teammates. This will be used to ensure that each student receives a deserved portion of their team's grade.

Additionally, students will be graded on traditional tests. These exams will be taken individually without outside resources.

The following is an outline of material presented in the text, Mechanics of Aircraft Structures by C.T. Sun

Preface to First Edition
Three main tasks of the text include:
 * 1. FEA modeling of problems and judging correctness of numerical results. Realizing new FEA codes have recently revolutionized the field of structural analysis in all mechanical applications.
 * 2. Fracture mechanics
 * 3. Composite properties in aircraft structures, brief study of laminates and the calculation of Young’s modulus derived from directional displacements as opposed to isotropic alloy mechanics for the purpose of describing torsion and bending problems.

In addition, Griffith’s Criterion is introduced for describing the relation between the strain energy release rate and crack extension. Also, buckling and post-buckling of bars and panels utilized in aircraft structures is introduced.

Preface Notes Second Edition modifications include material on the following topics: a Warping, Boundary Constraint Effects, Saint-Venant’s Principle, Shear Lag, Timoshenko Beam Theory, Plasticity and Fracture.



Introduction
Weight is the main consideration in aircraft structure design, unlike the structures of most civil engineering applications. Also, an aircraft’s structural design is modeled after the primary concern of the components, such as lift/drag considerations. Thus, limiting structural design options and resorting to material of the highest strength to weight ratios. Aircraft components typically follow the shell-like monocoque or stiffened shell semimonocoque structure. In the past, aluminum and titanium alloys have been the material of choice in aircraft design due to their high stiffness/weight and strength/weight ratios. Today, fiber-reinforced composites are replacing much of the weight in designs. Basic structural loads of an aircraft include axial, bending or torsional.

Axial Member
Axial members are employed to support loads applied, as the name would imply, in the axial direction.

Recall $$\sigma = E\epsilon\!$$ (Eqn 1.1 pg.2)

where


 * $$\sigma\!$$ is the "resulting uniaxial stress" from the load specified above (Mechanics of Aircraft Structures, C.T. Sun)
 * $$E\!$$ is the Young’s modulus and
 * $$\epsilon\!$$ is the normal strain in the loading direction and

Therefore $$F = A\sigma = EA\epsilon\!$$ (Eqn 1.2 pg.2)

where


 * $$F\!$$ is the total axial force and
 * $$A\!$$ is the cross sectional area

$$EA\!$$ is termed axial stiffness and is independent of the shape of the cross-section, rather determined by Young’s modulus and the cross-sectional area as illustrated in Figure 1. A large value of $$E\!$$ corresponds to high stiffness. For brittle materials the stress vs. strain curve is similar to that of Figure 2. For metals and other ductile materials, the stress vs. strain curve is similar to that of Figure 3. Channels are used in place of rods to increase bending stiffness.

Buckling strength is enhanced by adding lateral supports such as ribs and frames.



Shear Panel
These are thin sheets of material that carries in-plane shear loads. The following formula can be used to describe the shear force in the x-direction, $$V_x\!$$, for a shear panel under constant shear stress $$\tau\!$$ as illustrated in Figure 4. $$V_x = \tau ta = G\gamma ta\!$$ (Eqn 1.3 pg.4)

where


 * $$G\!$$ is shear modulus,
 * $$\gamma\!$$ is shear strain and
 * $$t\!$$ is thickness.



Similarly, curved plates may have their shear stress loads decomposed into $$V_x\!$$ and $$V_y\!$$ as follows and detailed in Figure 5.

$$V_x = \tau ta\!$$ (Eqn 1.4 pg.5)

$$V_y = \tau tb\!$$ (Eqn 1.5 pg.5)

and therefore the resultant force components have the following relationship

$$\frac{V_x}{V_y} = \frac{a}{b}\!$$ (pg.5)

Due to the ratio of geometric value in the $$x/y$$ directions and the corresponding shear stresses, a flat plate is the most efficient in providing shear force per unit of material.

Bending Member
These are members that carry a moment (Beams). Beams can also carry axial loads in tension or compression. Bending moments depend of beam deflection as shown below,

$$M = -EI\frac{d^2\omega}{dx^2}\!$$ (Eqn 1.6 pg.5)

where


 * $$M\!$$ is the bending moment,
 * $$I\!$$ is the moment of inertia and
 * $$\omega\!$$ is the previously mentioned beam deflection.

Where $$EA\!$$ was earlier termed as the axial stiffness, again the new term $$EI\!$$ is called the bending stiffness. Bending stresses lead to transverse stresses, however bending still plays the dominant role. Again, optimizing the cross-sections of beams is the key to increase bending stiffness. To utilize a materials full capacity, the cross-section should move material as far from the neutral axis as possible to take advantage of the linearity of the bending stress distribution over depth in the elastic range. An example of this is shown in Figure 6.

Torsion


Torque induced shear stress $$\tau\!$$ is also linearly distributed along the cross-section, only now it is distributed in the radial direction as shown in Figure 7.

$$T=GJ\theta\!$$ (Eqn 1.10 pg.7)

where


 * $$T\!$$ is the applied torque,
 * $$J\!$$ is the torsional constant and
 * $$\theta\!$$ is the angle of twist per unit length.

$$GJ\!$$ is known as the torsional stiffness and is dependent on the inner and outer radii for hollow cylinders, but generally depends upon the wall thickness. Thin walled structures are very efficient torsional members. To illustrate this, Figure 8 depicts three circular rods, each with equal cross-sectional areas, but with varied wall thicknesses.

It can be easily seen from the equation above that for a given material, the greater the torsional constant, $$J\!$$, the more torque, $$T\!$$, a beam can withstand. For a cylindrical beam, $$J\!$$ is defined by the following equation:

$$J = \frac{1}{2}\pi(b^4-a^4)\!$$ (Eqn 1.11 pg.8)

where


 * $$b\!$$ is the outer radius of the beam, and
 * $$a\!$$ is the inner radius of the beam.

Therefore,


 * $$J_{leftmost} = \frac{1}{2}\pi(1 cm)^4 = \frac{\pi}{2} cm^4\!$$
 * $$J_{middle} = \frac{1}{2}\pi((5.1 cm)^4-(5 cm)^4) = \frac{51.5\pi}{2} cm^4\!$$
 * $$J_{rightmost} = \frac{1}{2}\pi((10.05 cm)^4-(10 cm)^4) = \frac{201.5\pi}{2} cm^4\!$$

This helps show the advantages of thin walled beams in high torque operations, as the rightmost tube has a torsional constant which is two orders of magnitude greater than that of the solid member.

Load Transfer
Box beams due well to illustrate the beam and torsional members of aircraft substructure design. Loads generally are caused by air-pressure, landing gears, power-plants and seats, etc. Box beams are this sheets (shear panel) and longitudinal stringers (axial members) that distribute loads to major load-carrying members to avoid excessive deflection. Ribs collect all transverse loads from the stringers and transfer them to two wide-flange beams (spars).

Wing and Fuselage Structures


Main wing function is to carry air and power-plant loads to the fuselage. The wing itself acts as a beam and torsional member with an outer form designed by aerodynamic considerations. Spars are heavy beams that run span-wise to take transverse shear and bending loads. Wing ribs are planar structures that hold stringers to the desired contour and improve compressive ability of the wing. They are supported by span-wise spars. Subsonic aircraft have relatively thin skins and utilize spars and stringers as the main bending resistance. These wings may be composed of simple spars or a combination of spars and stringers. Supersonic wings on the other hand have thinner airfoils, at the same time requiring thicker skins to withstand high surface air-loads and improve bending resistance of the wing. To improve efficiency, stiffeners may be manufactured as integral parts of the wing. The fuselage must be designed to handle concentrated loads from wings and landing gears primarily. Also, payload and internal pressures must be supported. Again, stringers run along the length of the fuselage and rings maintain the shape of the fuselage and shorten the stringer length, increasing its bending resistance.

Aircraft Materials
These materials include metal alloys, such as steel, aluminum and titanium, and fiber-reinforced composites of either polymer, metal or ceramic matrices. Costs and properties of materials dictate their usage in design. Costs include manufacturing, maintaining and of course initial bulk. Properties vital to performance are density, strength, stiffness, durability, damage tolerance and corrosion resistance. Steel alloys are denser and corrosion prone so they are used for highly loaded critical points and must be coated. Aluminum alloys range from higher strength to higher toughness; yet both are light-weight and serve vital tasks at different sections of the aircraft. Titanium is lighter and stronger than steel but much more expensive and therefore used mostly in military aircraft. They also withstand higher temperatures than aluminum (350o F versus 1000o F).


 * Materials with high stiffness and high strength - steel alloys, titanium alloys, and aluminum alloys


 * Materials with high stiffness and low toughness - glass (toughness is ability to resist fracture)


 * Materials with low stiffness and high toughness - plastic, nylon, aluminum


 * Materials with high stiffness and high toughness - composites

Fiber-reinforced composites usually employ unidirectional fibers of high tensile strength in a matrix of polymer, metal or ceramic material. These sections form thin laminae with a stacking pattern of different fiber orientations to produce laminates with excellent material properties and multidirectional load capabilities. Ceramic composites in particular have excellent heat resistance, fatigue life, damage tolerance and corrosion resistance. One difference between fiber reinforced composites and metal matrix composites is that the metal matrix composite has a metal matrix, with some type of reinforcement. The fiber reinforced composite can have several different types of matrices.

Contributing Team Members
The following students contributed to this report:

Scott Chastain Eas4200c.f08.aero6.chastain 19:52, 12 September 2008 (UTC)

Marlana Behnke Eas4200c.f08.aero6.behnke 19:57, 12 September 2008 (UTC)

Matt Inman Eas4200c.f08.aero6.inman 20:18, 12 September 2008 (UTC)

Eduardo J. Villalba Eas4200c.f08.aero6.villalba 20:40, 12 September 2008 (UTC)

Felipe Ortega Eas4200c.f08.aero6.ortega 18:51, 19 September 2008 (UTC)

References

Figure 2- http://en.wikipedia.org/wiki/Image:Stress_v_strain_A36_2.png

Figure 3- http://en.wikipedia.org/wiki/Image:Stress_v_strain_brittle_2.png nq