User:Eas4200c.f08.blue.a/hw1lecnotes

Contact Information
Instructor: Dr. L. Vu-Quoc


 * Telephone: 392-6227
 * Email: vu-quoc@ufl.edu
 * Course Website: EGM4200c Aerospace Structures
 * Office: NEB 135 (New Engineering Building)
 * Classroom: MAE-A 303
 * Class Time: MWF, period 4 (10:40 am - 11:30 am)
 * Office Hours: MWF, period 9 (4:05 pm - 4:55 p)

Brief Summary of Course Objectives
The objective of the course "Aerospace Structures" is to establish an understanding of the mechanics of aerospace structures. With this, the student will develop the ability to formulate problems pertaining to aerospace structures and judge the correctness of the solutions formulated. In developing these methods, old methods will be avoided and partial differential equations will be solved by new methods like the finite element method.

Informative Links
| Aerospace Engineering

| Aerospace Structures

| MAE at The University of Florida

The use of Wikipedia for Homework Reports
Setting up wikipedia accounts: Format: Username: EAS4200c.f08.team_name.last_name Password: Up to user

Special Note: Class wikipedia accounts should NOT be used for editing pages other than the class assigned pages. Failure to follow these instructions could lead to punishment.

Wikipedia will be used for the submission of homework reports. The reports will consist of 2 parts. The first being lecture note re-writes and the second part being homework problems.

Students are expected to attend all lectures and take good, accurate notes. The notes should then be compiled with the other group members and posted on the group's wikipedia page. On the page, the notes should be organized according to subject, not lecture number. It is important not to simply re-write the notes but also add to them. The instructor will make suggestions on where to contribute to the notes. Examples of contributions include, explanations of phenomena and derivations of equations.

The Vision for EAS 4200c
The Big Picture

Wikipedia is part of the Mediawiki software. The Mediawiki software is used in many different programs and websites and allows for collaboration from all users from any location in the world. Learning the use of this software will not only allow students to succeed in this class, but also make other contributions on other Mediawiki projects.

Confidentiality

Because some students have expressed concern over having their last names in the wikipedia usernames, groups have the option of using letters (i.e. a,b,c,d,e) to identify their individual userpages.

Method of Work

The new method of work is as follows: 31% for group homewoek reports, and 23% for each of three exams. Also during each exam, students will fill out group evaluations about their fellow group members. These will factor into the fonal homework grades.

Old Approach

The old approach which didn't utilize the Mediawiki software had a different grading scale. The old system had homework worth 10%, and each of the exams are worth 30%. For now, we will try the new method, where homework is worth 31%, and each exam is worth 23%. If at any point during the year, a majority of students feel like the new method doesn't work, we can revert to the old method. We will give it, "the old college try."

E-Learning

Dr. Vu-Quoc has decided not to use the e-learning system provided by UF for this class. He is not familiar with the software and feels that the Mediawiki software is perfect for the course plan.

Collaboration

This class involves collaboration among several groups. The first is collaboration among the students in the group. Next, is the collaboration among the class groups. Lastly is collaboration among all of the groups and the professor. By making all the class notes availiable to everyone, it will enhance the learning experience.

Introduction to Aerospace structures
The goals of aerospace structures is to design aircraft that are very light yet also strong. this is accomplished through the use of materials that have a high stiffness, high strength, and lightweight.

The stiffness of a material is determined by $$E$$ The young's modulus of the material which is relates the stress and strain on the material through hooke's law $$\sigma=E*\epsilon$$ hooke's law is a linear equation and the Young's modulus is the slope of the equation. This means for the higher the Young's modulus the faster the stress will increase meaning little deformation will be able to occur in the material before the yield stress is reached.

The strength of a material is determined by $$\sigma_y $$, $$ \sigma_u$$ which are the yield stress and the ultimate stress respectively.

The higher the stresses that the material is able to handle before yielding the stronger the material.

Toughness is also considered in materials because it is the ability of the material to resist fracturing. Which means that a material with a high toughness would be able to resist fracturing very well. An example of a very stiff material but has a very low toughness would be glass.

Two aspects of aircraft design are the geometry and materials used. The geometry for the aircraft body is constructed by a thin wall shell design instead of an internal truss design. Aerodynamics also play a larger role in the geometry of the aircraft. Materials that are used need to be used to their full capacity to avoid more weight than necessary in the design. Also when able replace aluminum and titanium with fiber reinforced composites which will reduce weight by approximately thirty to forty percent.

Book pg 1-3

Problem 1.1


The problem is to find the optimal $$b/a$$ in a thin walled beam to maximize the load-bearing capacity of a beam while assuming that the moment (M) is equal to the torsion (T).This is done by optimizing the cross section that carries the maximum bending moment and the maximum torsion.

Finding the Solution begins with the following assumptions: 1. The wall thickness is much less than the width and height of the beam (t<<a, t<<b)  2. The Moment on the beam is equal to the Torsion on the beam (M=T)  3. The allowable stress is equal to twice the allowable shear ($$\sigma _{allowable}= 2 \tau _{allowable}$$)  4. The beam is rectangular in shape, ie. the perimeter of the beam face (L) is equal to twice the sum of length a and b (L=2(a+b)).

Next, the shear stress ($$\tau$$) due to Torsion (T) is found. First, the total Torsion is found by separating the beam into four different elements, finding the Torsion in each, and adding them together.



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

Knowing that $$\tau =\frac{\nu }{t}$$, we are able to solve for Torsion in each element of the area of the beam.

$$T_{AB}=\frac{b}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{BC}=\frac{a}{2}\times (\nu \times a)=\frac{1}{2}\tau abt$$

$$T_{CD}=T_{BC}$$

$$T_{DA}=T_{CD}$$

$$T=2 \tau abt$$

$$\tau =\frac{T}{2abt}$$

We consider two cases for the solution.

Case1

In Case 1 we assume that the bending normal stress ($$\sigma $$) reaches the allowable stress ($$\sigma _{allowable}$$). The goal is to verify that the shear stress in the beam ($$\tau $$) is less than or equal to the allowable shear stress ($$\tau _{allowable}$$). The first step in this case is to define the bending moment as follows:

$$\sigma =\frac{Mz}{I}$$

Where M is the bending moment, z is the ordinate of a point adn I is the second area moment of inertia defined as: $$I=\int \int_{A}^{}{z^{2}dydz}$$

Because the shear stress is greatest at the top and bottom of the beam section (where z = b/2), the equation of shear stress can be rewritten as follows: $$\sigma = \frac{Mz}{I}  \Rightarrow   M=\frac{2I\sigma\ _{allowable}}{b} $$

Thus, it is necessary to find the value of I in terms of b (recalling that a can be written as a function of b: a=L/2-b). From the problem parameters we know that for this instance, Mmax = Tmax. Because Mmax has been determined, it must only be put into the relation:

$$\tau _{max}=\frac{T^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}$$

Now check the values of $$\tau _{max}$$ with the value of $$\tau _{allowed}$$.If $$\tau _{max}$$ is larger than $$\tau _{allowable}$$ the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$would be acceptable. But $$\tau _{max}$$ is smaller than $$\tau _{allowable}$$, then the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ would be unacceptable

Case 2

For this case we assume that the shear stress is eqaul to the maximum allowable shear stress in the section. Then we will test to see if the normal stress falls within acceptable tolerances (notably, less than $$\sigma _{max}$$)

By the equation given earlier in the problem the value of T becomes:

$$T=(2t\tau _{allowable})(ab)$$

Where the dimensions of the rectangle ab are equal to one another, hence a square. Therefore if the total perimeter of the square is L and all sides are equal, both a and b are equal to L/4. Plugging this quantity into the previous equation and using the given relation that M=T, we have:

$$M^{(2)}_{max}=\frac{1}{8}tL^{2}\tau _{allowable}$$

Using this relation and the previously defined equation for normal stress:

$$\sigma ^{(2)}_{max}=\frac{M^{(2)}_{max}b^{(2)}}{2I^{(2)}}$$

We can find $$\sigma ^{(2)}_{max}$$ and compare it with the allowable normal stress. If the maximum normal stress is less than the allowable shear stress, then the ratio of b to a for case 2 is acceptable. But, if it is greater, then that ratio is not correct.

Solving Case 1

First the moment of inertia, I, must be computed. This is done by summing the moments of inertia of each of the four parts of the section. In order to do this, the Parallel axis theorem must be applied. Solving for the horizontal axis yields:

$$I=\frac{tb^{2}}{6}(3a+b)$$

And with the relation a=L/2-b, the plot of f(b)=I/b is shown to the right.

Taking the derivative of the function f(b) yields the maximum value of the curve to be b(1)=3L/8. Thus the ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ is equal to 3 and the value of a(1) is found to be L/8. Solving for $$\tau _{max}$$ with the values of a(1) and b(1) yield:

$$\tau _{max}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{32M^{(1)}_{max}}{3tL^{2}}$$

but because this larger than half of $$\tau _{allowable}$$ this case is not possible

Team Members That Contributed to This Report
Mark Barry Eas4200c.f08.blue.a 18:20, 18 September 2008 (UTC)

Clay Robertson Eas4200c.f08.blue.f 05:11, 19 September 2008 (UTC)

Christian Garabaya Eas4200c.f08.blue.d 05:24, 19 September 2008 (UTC)

Jeffrey Shea Eas4200c.f08.blue.e 12:13, 19 September 2008 (UTC)

Marc Pelletier Eas4200c.f08.blue.b EAS4200c.f08.blue.b 15:24, 19 September 2008 (UTC)

Thomas Sexton EAS4200c.f08.blue.c 16:16, 19 September 2008 (UTC)