User:Eml4500.f08.group/AAnotes

Mon, 25 Aug 08
Class Website http://clesm.mae.ufl.edu/~vql/courses/fead/2008.fall/

FEM wide spread use in all reas of engineering applied math, heologival esplorations of oil reseriors etc

Password for mediawiki: fall08.wiki

Wed, 27 Aug 08
Deadline for the team structure: - Wed, 3 Sep 08 Each team, (5 or 6 students) Submit a paper:


 * Team Name: 4 Char preferred, max 6 char.
 * member names + UFID Numbers
 * Wikipedia Class usernames

Put your contribution is collapsible table

Vision - Professor smoking the Wiki Pipe

 * Big Picture re Wikipedia
 * textbooks, Wikiversity
 * MIT's OpenCourseWare


 * MediaWiki
 * Wikiversity - MIT's OpenCourseWare
 * Collaboration between students in a team, collaboration of whole class, collaboration of prof and class

Old approach
10% HW, 30% + 30% + 30% Exams

Trusses, Matrix Method (Chapter 4)
Also MIT's Open Course Ware

Class Wiki Pages Truss with 2 elastic (deformable) bars

Both "Bases" Constrained, no displacement

Global FBD
Whole Structure

4 unknown reactions, 3 equations of equilibria, statically indeterminate

The look at Free body diagrams for each element,

(n) Global Node number n [n] Local Node number n fi = ith internal force of element e: 1 = 1,2,3,4 : e = 1, 2

=Fri, 5 Sep 08=

Reading
READ: Chap 4 (Trusses, beams, frames)
 * Sections
 * 1.1.1 Plane Truss Elem
 * 1.2 Assembly of elem eqs
 * ex 1.4 Five-bar truss
 * Elem Solutions and model ralidity
 * 1.4.1 Plane Truss element

Steps to solve simple truss system
Steps to solve simple truss system described on Wed, 3 Sep 08. (Recipe)

Global picture
(description) at structure level
 * Global degrees of freedom (disp coeffs)
 * Global Forces

Actually the displacement degrees of freedom are partitioned into Simularly for the global forces
 * A known part, eq, fixed defs, constraints
 * an unknown part, solved using FEA
 * a know part: applied force
 * an unknown part: reactions

Element picture
either in global coord system on in local coord system
 * element degrees of freedom
 * element forces

Global Force Displacement relationship

 * Elemenet Stiffness matrix in global coord
 * element force matrices in global coord.
 * Assembly of element stiffness matrix and element force matrix into a global Force Displacement relationship

Elimination of known degrees of freedom
Elimination of known degrees of freedom to reduce the global Force Displacement Relationship (stiffness matrix non-singular => invertible)


 * m = number of unknown displacement degrees of freedom
 * n = number of known and unknown and displacement degrees of freedom

Kbar non singular => Kbar^-1   (Kbar invertible)

Compute Element forces
Compute Element forces from now known dbar => element stresses

Compute Reactions
Compute reactions (unknown forces)

Example
Take a specific example to see how the recipe works (no justification yet)

Element Length - L(1) = 4, L(2) = 2

Youngs Modulus - E(1) = 3, E(2) = 5

Cross Section Area - A(1) = 1, A(2) = 2

=Mon, 8 Sep 08= 2.) Element Picture

3.) Global Force Displacement at element level



k is the element stiffness matrix for element e

d is the the element displacement matrix

f is the element force matrix

Matrix p 225 book, 2nd equation form the bottom





axial stiffness of bar elem "e". e = 1,2

l^(e), m^(e) = director cosines of ~x axis (goes from [1] to [2]) with respect to global (x,y) coords

=Wed, 10 Sep 08= [7] Model 2-bar truss system (cont'd)

Note: The director cosines are the components of i^~-> (unit vector along x axis, with respect to basics (i^->,j^->)

Element 1






Observations

 * 1) You only need to compute 3 numbers. Other coefficients have the same absolute values, just differ by (+) or (-).
 * 2) Symmetry [[Image:FEAeq-7-3.JPG]]

transpose results in (note the dangle symmetry



Element 2


k(1,1) = 2.5

Observation

 * 1) The absolute values of all coefficients are the same. You only need to compute 1 coefficient
 * 2) Symmetry [[Image:FEAeq-7-3.JPG]]

Element force displacement relationship:





Global Force Displacement relationship



n = 6, this is because there are 6 outside forces acting on the system. 2 in each of the X and Y directions at the 3 points of the truss.