User:Eml4500.f08.team.guan/fixed1017

 HW Redo: Further explanation for the math is added. The third motivation for the PVW is added. Initial derivation of the PVW is added.

Equilibrium of Node 2:
Now, the sum of the forces in the x and the y directions are put into equation form in the following equations. Then, using the force displacement relationship, this forces in the equation are replaced with the product of the stiffness and the corresponding displacement degree of freedom.


 * Eq (1):

$$f_3^{(1)}+f_1^{(2)}=0 \quad (1)$$


 * Eq (2):

$$f_4^{(1)}+f_2^{(2)}=\underline{P}\quad (2)$$


 * From p 8-3:

$$d_3=d_3^{(1)}=d_1^{(2)}$$ $$d_4=d_4^{(1)}=d_4^{(2)}$$

Assembly Rows 3 & 4 From Eq (1) & (2) into:
Recall that the global stiffness matrix has the following values. Replacing the forces in equation one with the product of the stiffness and displacement degrees of freedom results in the following new equation one. Thus, this new equation one is consequently the third row of the global stiffnes matrix on 9.1. Also see 8.4 for the global force displacement relationship. The new equation two is similarly derived.

$$\mathbf{K}= \begin{bmatrix} k_{11}^{(1)} & k_{12}^{(1)} & k_{13}^{(1)} & k_{14}^{(1)} & 0 & 0\\k_{21}^{(1)} & k_{22}^{(1)} & k_{23}^{(1)} & k_{24}^{(1)} & 0 & 0\\k_{31}^{(1)} & k_{32}^{(1)} & (k_{33}^{(1)}+k_{11}^{(2)}) & (k_{34}^{(1)}+k_{12}^{(2)}) & k_{13}^{(2)} & k_{14}^{(2)}\\k_{41}^{(1)} & k_{42}^{(1)} & (k_{43}^{(1)}+k_{21}^{(2)}) & (k_{44}^{(1)}+k_{22}^{(2)}) & k_{23}^{(2)} & k_{24}^{(2)}\\0 & 0 & k_{31}^{(2)} & k_{32}^{(2)} & k_{33}^{(2)} & k_{34}^{(2)}\\0 & 0 & k_{41}^{(2)} & k_{42}^{(2)} & k_{43}^{(2)} & k_{44}^{(2)}\end{bmatrix}$$

$$\left [ k_{31}^{(1)}*d_1+k_{32}^{(1)}*d_2+k_{33}^{(1)}*d_3+k_{34}^{(1)}*d_4 \right ]+\left [ k_{11}^{(2)}*d_3+k_{12}^{(2)}*d_4+k_{13}^{(2)}*d_5+k_{14}^{(2)}*d_6 \right ]=0\quad (1)$$

$$\left [ k_{41}^{(1)}*d_1+k_{42}^{(1)}*d_2+k_{43}^{(1)}*d_3+k_{44}^{(1)}*d_4 \right ]+\left [ k_{21}^{(2)}*d_3+k_{22}^{(2)}*d_4+k_{23}^{(2)}*d_5+k_{24}^{(2)}*d_6 \right ]=\underline{P}\quad (2)$$

Now, an expression can be written to describe the assembly of a general (an arbitrary number of elements) global stiffness matrix using general elemental stiffness matrices. Assembly of $$\underline{k^{(e)}}$$, e = 1, ....., nel (number of elements), into global stiffness matrix K is as follows. It is necessary to denote a symbol other than $$\sum$$ to allow the below assembly expression to be possible and true. This is because the global stiffness matrix and the elemental stiffness matrices are of different dimensions where the dimensions of the global stiffness matrix can be millions by millions. Thus, simple matrix addition cannot be utilized here and another symbol other than $$\sum$$ must be used. Let "A" be this designated assembly operator.

Principle of Virtual Work (PVW)
There are three motivations for applying the principle of virtual work. 1) (p 10-1): Elimination of rows corresponding boundary conditions to obtain $$\underline{K}$$.

$$\underline{K} = \begin{bmatrix}K_{33} & K_{34}\\ K_{43} & K_{44}\\ \end{bmatrix}$$

2)(p 12-1): Element Force-Displacement Relationship described by axial forces and displacements and the transformation matrix T.

$$ \mathbf{\underset{2 \times 1}q^{(e)}}=\mathbf{\underset{2 \times 4}T^{(e)}}\mathbf{\underset{4 \times 1}d^{(e)}} $$

This leads to the following relationship:

$$	\underbrace{\begin{pmatrix}

q^{(e)}_1 \\

q^{(e)}_2 \end{pmatrix}}_{\underset{2 \times 1}q^{(e)}} = \underbrace{\begin{bmatrix}

l^{(e)} & m^{(e)} & 0 & 0 \\

0 & 0 & l^{(e)} & m^{(e)} \\ \end{bmatrix}}_{\underset{2 \times 4}T^{(e)}}\underbrace{\begin{pmatrix}

d^{(e)}_1 \\ d^{(e)}_2 \\ d^{(e)}_3 \\ d^{(e)}_4 \end{pmatrix}}_{\underset{4 \times 1}d^{(e)}}$$

$$\mathbf k^{e} = \mathbf {T^{(e)^T} \hat k^{(e)} T^{(e)}}$$ 3) Deriving the finite element method for partial differential equations (PDEs). Let's begin by stating the force displacement relationship for a bar of a spring. $$\displaystyle Kd=F, Kd-F=0...(3)$$ $$\displaystyle w(Kd-F)=0$$ for all w $$\displaystyle ...(4)$$ Equation four is the "weak form" which is the principle of virtual work. The proof is as follows. a) The fact that equation 3 implies equation 4 is trivial. One can just plug in equation 3 into 4 and get zero. b) However equation 4 implying equation 3is not trivial since equation 4 is valid for all w. You can select any arbitrary value say w=1 then equation 4 becomes 1(Kd-F)=0. Therefore, equation 3 is obtained.