User:Eml4500.f08.ateam.nobrega/Lecture Notes Homework 1

Introduction to EML4500 Finite Element Analysis
The start of EML4500 was an introduction of the course detailing the schedule, the grading system, and the method for submitting work in the class. Groups of five or six students were to be created to work on homework and submit homework reports. These reports were originally going to be submitted on Wikipedia, but are now to be submitted on Wikiversity. Dr. Vu-Quoc also introduced the concept of collaborative learning and its benefits when juxtaposed with traditional teaching/learning methods. Each team is responsible for summarizing each lecture and identifying confusing or ambiguous points discussed. All of the teams' input will be used by Dr. Vo-Quoc as feedback and to address any underlying issues if necessary. Using Wikiversity would enable a sharing of knowledge between different teams enabling the class to be better prepared for the exams and encouraging discussions of course content. This would be accomplished by the posting of the top three homework reports on the class main page, making the correct answers to the homework available to the whole class with the material presented in a clear and concise manner. The top three teams would be given extra credit, but if any flaws were discovered in their reports after posting some of the extra credit could be taken away and given to those who found the flaws.

Using Wikiversity
Instructions were given on how to register and use Wikipedia (Wikiversity). It was explained to the class how to create an account and access one's user page as well as the class page. To do this the students were instructed to go to the Wikiversity main page and select

log in/create account

Next the user would need to type in the appropriate user name and create a password. The user names were to be created as follows:

eml4500.f08.teamname.lastname.first initial

Adding the first initial at the end of the user name was optional. We were then taught how to create links in our user page. Links are created by typing user:eml4500.f08.teamname.lastname.first initial/link|Link Name into ______________. This allows the user to create multiple pages within their user page or to link to a group member's user page or the class page.

Following this discussion it was explained in more detail what is expected of our homework reports, including content and what is expected of each group member. As a group member one is expected to work on his or her part and post in his or her user page. Once one finishes their contribution they are to post it on the group leader's user page where it will be compiled in the final report. This allows the Teaching Assistants and the Professor to possibly view each group members contribution to the homework report as well as the times of each submission. To account for any disparity between individual group member's contribution the group will rate each member's productivity on a scale from 0 to 8. The ratings will be tallied and a percentage will be taken based on the best possible rating one can receive. This percentage is then multiplied with the group's homework report grade and that member's individual homework grade is determined.

The advantages of using Wikiversity and Wikimedia followed, ending this series of notes with a comparison to the older more traditional method of structuring a course. It was explained to the class that at any point in the semester, if the majority of the class was in agreement, it would be possible to switch back to the more traditional method and grading scale.

Trusses, Beams and Frames (Ch4)
Trusses are structural systems arranged to resist primarily axial forces. Beams are long slender members that are commonly subjected to normal loading with respect to their longitudinal axis; as a result, they must withstand shear and bending forces. Frames can be viewed as a hybrid between trusses and beams because they are meant to resist bending and axial forces. Consider a truss composed of two elastic (deformable) bars as shown in Figure 1. The bars are fixed (constrained to zero displacement) at the two ends shown and a known force P is applied at the joint of the two bars. By inspection one realizes that there are four reaction forces, one in the x-direction and one in the y-direction of each fixed end.



Another method may be appropriate to solve this problem. First, inspect the system as a whole by drawing a global force body diagram (FBD). This diagram is featured below in Figure 2 and the FBDs of bar elements one and two are represented in Figures 3 and 4 respectively.



Then analyze each individual component by drawing the individual bar element FBD. In the Global FBD each node number is placed inside of a circle, for the FBD of the individual bar elements the node number is distinguished by a local node number that is placed inside of a square. The number may or may not directly correspond to the Global node number that it is referencing. For example, with bar element one in figure 3 below the Global node number one is also the local node number one, but depending on the user's preference the local node number for global node one could be two and the local node number for two could be one. This is also apparent with bar element two in figure 4 below as Global node number two is local node number one and Global node number three is local node number two. For the forces shown in the FBD's of the two bar elements the correct nomenclature is a capital F with a superscript of the bar element number in parenthesis and a subscript degree of freedom. It is correct to start with the x-direction and then continue with the y-direction at each node when labeling the degree of freedom.



For bar elements one and two the displacements directly correspond to the forces and are to follow the same strategies listed above when labeling the diagram. Instead of a capital F being used, for the displacements a lower case d will occupy its space.

The next big step is determining the force displacement (FD) relationship.

If you recall from previous classes the spring force equation is:

$$F=kd$$

This equation represents the force-distance relationship for a one-dimensional spring.

For a one-dimensional spring with 2 free ends the force-displacement relationship is more complex than the previous equation and can be determined with the use of matrices.

The force and displacement of the spring is shown in the image below:



When placing the equation for the spring above into matrix form one gets,



The equation above is a simple spring force equation for a spring with two nodes. The left side of the equation represents the spring forces at node one and two in a 2x1 matrix. The right hand side of the equation consists of the spring constant and the displacement of each node. The spring constant is listed in a 2x2 matrix while the displacement belongs in a 2x1 matrix.

Case 1: For the figure above with the observer is seated at node one

$$F$$2$$= k*$$$$(d$$2 $$- d$$1$$)$$

Case 2: For the figure above with the observer seated at node two or at equilibrium where F1 + F2 = 0

$$F$$1$$ = -F$$2$$= k*(d$$1$$ - d$$2$$)$$

Instructions (Recipe): Simple Trusses
This lecture given by Professor Vu-Quoc was a “recipe” for the steps to solve a simple truss system. The steps for solving a simple truss system are as follows:

Step 1 describes the Global Picture at the structure level. Identify the Global Degrees of Freedom (dof). (The displacement dofs) The displacement dofs are divided into a know part which is the fixed dof constraint, and unknown part which is solved for using the Finite Element Method. Next Identify the Global Forces. Similar the Global dofs the Global Forces are also divided into two parts. The know parts are the applied forces, and the unknown are the reaction forces.

Step 2 describes the Element Picture. First break the truss down into individual elements and identify element dofs, and element forces. When identifying the element dofs and forces you may either use the global or a local element coordinate system to make computation easier.

Step 3 defines the Global Force Displacement Relation. Define the element stiffness matrix K in the global coordinate system. Define The element force matrix F in the global coordinate system. The final step is the assembly of the element stiffness matrix and element force matrix into the global force displacement relation. The global force displacement relation is given by the equation: Kd=F where K is a(nxn) matrix, d is a (nx1) matrix, and F is a (nx1) matrix.It is assumed that the stiffness matrix is non-singular and invertible.

Step 4 the elimination of the known dofs to reduce the global Force Displacement Relation. Because the stiffness matrix K is non-singular and invertible the number of unknown displacement dofs are less than the number of know displacement dofs. The equation above in Step 3 is then solved for d by inverting K and multiplying by F resulting in d=K(inverse)F

Step 5 compute the element Forces from the now known ‘’’d’’’ element stresses.

Step 6 compute the unknown reaction Forces from the now known element forces and element stresses.

For further explanation please see the illustrated example below.

Example Problem:

The example below will showcase the steps described in the recipe above to solving simple truss problems. Trusses are composed of elements that only allow for axial loads. The elements are connected by nodes or joints. This problem has two elements and three nodes where each node has two degrees of freedom. The problem indicates an applied load P is present at node two. After solving the problem it would be possible to quantify the deflections and force reactions at each node.

Data:

Element Length: *L(1) = 4 *L(2) = 2

Young's Modulus: *E(1) = 3 *E(2) = 5

Cross Sectional Area: *A(1) = 1 *A(2) = 2

Inclination Angle: *Θ(1) = 30o *Θ(2) = 45o

Step 1: Global Picture

Labeling : Global degrees of freedom (dofs) and forces

The numbers inscribed in the green circles refer to a node value n. In the example described above the node values range from one to three. When numbering the displacement degrees of freedom (d1, . . . d6), you want to start at node one and with the x displacement. This displacement degree of freedom would be labeled ( d1 ). This would be followed by the y displacement at node one; labeled ( d2 ). Next would be node two, again starting with the x displacement; labeled ( d3 ). Followed again by the y displacement; labeled ( d4 ). One would continue this process until all the nodes are labeled with their corresponding degrees of freedom. The same systematic order would be followed to label the global forces ( F1, . . . F6 ). Refer to the image above for further explanation on systematic labeling. These displacements and forces are then put into a matrix format as seen in the image below.