User:Eml4500.f08.ateam.rivero/w4

Meeting 23: October 17, 2008
Recall Equations:

f_{3}^{(1)} + f_{1}^{(2)} = 0 $$
 * Equation 1: $$

$$ [k_{31}^{(1)}*d_{1}^{(1)} + k_{32}^{(1)}*d_{2}^{(1)} + k_{33}^{(1)}*d_{3}^{(1)} + k_{34}^{(1)}*d_{4}^{(1)}] +         [k_{11}^{(1)}*d_{1}^{(2)} +            k_{12}^{(1)}*d_{2}^{(2)} + k_{13}^{(1)}*d_{3}^{(2)} + k_{14}^{(1)}*d_{4}^{(2)}] = 0 $$
 * Equation 1: Expanded

$$ [k_{31}^{(1)}*d_{1} + k_{32}^{(1)}*d_{2} + k_{33}^{(1)}*d_{3} + k_{34}^{(1)}*d_{4}] + [k_{11}^{(1)}*d_{3} + k_{12}^{(1)}*d_{4} + k_{13}^{(1)}*d_{5} + k_{14}^{(1)}*d_{6}] = 0 $$
 * Equation 1: Expanded using global degrees of freedom

f_{4}^{(1)} + f_{2}^{(2)} = P $$
 * Equation 2: $$

$$ [k_{41}^{(1)}*d_{1}^{(1)} + k_{42}^{(1)}*d_{2}^{(1)} + k_{43}^{(1)}*d_{3}^{(1)} + k_{44}^{(1)}*d_{4}^{(1)}] + [k_{21}^{(1)}*d_{1}^{(2)} +           k_{22}^{(1)}*d_{2}^{(2)} + k_{23}^{(1)}*d_{3}^{(2)} + k_{24}^{(1)}*d_{4}^{(2)}] = P $$
 * Equation 2: Expanded

$$ [k_{41}^{(1)}*d_{1} + k_{42}^{(1)}*d_{2} + k_{43}^{(1)}*d_{3} + k_{44}^{(1)}*d_{4}] + [k_{21}^{(1)}*d_{3} + k_{22}^{(1)}*d_{4} + k_{23}^{(1)}*d_{5} + k_{24}^{(1)}*d_{6}] = P $$
 * Equation 2: Expanded using global degrees of freedom

Assembly of   $$  \displaystyle \mathbf     k^{(e)} $$, e = 1,. . ., nel (number of elements) into the global stiffness matrix

$$\displaystyle \mathbf K = \overset{e=nel}{\underset{e=1}{\mathbb A}} \mathbf k^{(e)}$$ $$\displaystyle \overset{e=nel}{\underset{e=1}{\mathbb A}} $$ = Assembly Operator n : total number of global degrees of freedom before eliminating the boundry conditions ned : number of element degrees of freedom ned << n : ned is very small when compared to n

Principle of Virtual Work
Motivation:


 * The principle of virtual work allows us to eliminate the corresponding rows to the boundry conditions to obtain the global stiffness matrix K. Reducing the global stiffness matrix from a (6x2) matrix to a (2x2) matrix.

q^{(e)} = T^{(e)}\cdot d^{(e)} $$



k^{(e)} = T^{(e)T}\cdot k^{(e)}\cdot T^{(e)} $$


 * The ultimate goal is that the principal of virtual work allows us to derive the Finite Element Method for Partial Differential Equations which are heavily used in the world of Heat Transfer, Vibrations, Finite Element Analysis Commercial Codes and many more.

Force Displacement Relation for a bar or spring is expressed in the form $$ kd=F $$, which implies

$$ kd-F=0 $$ Equation (3) $$ w(kd-F)=0 $$ for all w Equation (4) Proof: (3) => (4) Trivial (4) => (3) Not Trivial The reason why going from equation 3 to 4 is trivial is because since equation 4 is valid for all w, select w = 1 and then equation 4 becomes.... $$ 1(kd-F) = 0 $$ which is equal to equation 3