Strumenti Utente



scienza_costruzioni:fem:vincoli_interni

Vincoli interni

Supponiamo i sia l'indice di uno dei gradi svincolati. In questo caso possiamo scrivere

$$ \begin{Bmatrix} f_{L,1} \\\\ \vdots \\\\ f_{L,i-1} \\\\ 0 \\\\ f_{L,i+1} \\\\ \vdots \\\\ f_{L,n} \end{Bmatrix} = \begin{bmatrix} k_{L,1,1} & \dots & k_{L,1,i-1} & k_{L,1,i} & k_{L,1,i+1} & \dots & k_{L,1,n}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{L,i-1,1} & \dots & k_{L,i-1,i-1} & k_{L,i-1,i} & k_{L,i-1,i+1} & \dots & k_{L,i-1,n}\\\\ k_{L,i,1} & \dots & k_{L,i,i-1} & k_{L,i,i} & k_{L,i,i+1} & \dots & k_{L,i,n}\\\\ k_{L,i+1,1} & \dots & k_{L,i+1,i-1} & k_{L,i+1,i} & k_{L,i+1,i+1} & \dots & k_{L,i+1,n}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{L,n,1} & \dots & k_{L,n,i-1} & k_{L,n,i} & k_{L,n,i+1} & \dots & k_{L,n,n} \end{bmatrix} \begin{Bmatrix} \eta_{L,1} \\\\ \vdots \\\\ \eta_{L,i-1}\\\\ \eta_{L,i} \\\\ \eta_{L,i+1}\\\\ \vdots \\\\ \eta_{L,n} \end{Bmatrix} + \begin{Bmatrix} f_{L,0,1} \\\\ \vdots \\\\ f_{L,0,i-1} \\\\ f_{L,0,i} \\\\ f_{L,0,i+1} \\\\ \vdots \\\\ f_{L,0,n} \end{Bmatrix}$$

Facciamo in modo che il termine i-esimo di ciascuna riga della matrice sia uguale all'unità, dividendo quindi la riga j-esima per $k_{L,j,i}$

$$ \begin{Bmatrix} \frac{f_1}{k_{1,i}} \\\\ \vdots \\\\ \frac{f_{i-1}}{k_{i-1,i}} \\\\ 0 \\\\ \frac{f_{i+1}}{k_{i+1,i}} \\\\ \vdots \\\\ \frac{f_n}{k_{n,i}} \end{Bmatrix} = \begin{bmatrix} \frac{k_{1,1}}{k_{1,i}} & \dots & \frac{k_{1,i-1}}{k_{1,i}} & 1 & \frac{k_{1,i+1}}{k_{1,i}} & \dots & \frac{k_{1,n}}{k_{1,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ \frac{k_{i-1,1}}{k_{i-1,i}} & \dots & \frac{k_{i-1,i-1}}{k_{i-1,i}} & 1 & \frac{k_{i-1,i+1}}{k_{i-1,i}} & \dots & \frac{k_{i-1,n}}{k_{i-1,i}}\\\\ \frac{k_{i,1}}{k_{i,i}} & \dots & \frac{k_{i,i-1}}{k_{i,i}} & 1 & \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{i,n}}{k_{i,i}}\\\\ \frac{k_{i+1,1}}{k_{i+1,i}} & \dots & \frac{k_{i+1,i-1}}{k_{i+1,i}} & 1 & \frac{k_{i+1,i+1}}{k_{i+1,i}} & \dots & \frac{k_{i+1,n}}{k_{i+1,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ \frac{k_{n,1}}{k_{n,i}} & \dots & \frac{k_{n,i-1}}{k_{n,i}} & 1 & \frac{k_{n,i+1}}{k_{n,i}} & \dots & \frac{k_{n,n}}{k_{n,i}} \end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1}\\\\ \eta_{i} \\\\ \eta_{i+1}\\\\ \vdots \\\\ \eta_n \end{Bmatrix} + \begin{Bmatrix} \frac{f_{0,1}}{k_{1,i}} \\\\ \vdots \\\\ \frac{f_{0,i-1}}{k_{i-1,i}} \\\\ \frac{f_{0,i}}{k_{i,i}} \\\\ \frac{f_{0,i+1}}{k_{i+1,i}} \\\\ \vdots \\\\ \frac{f_{0,n}}{k_{n,i}} \end{Bmatrix}$$

Sottraendo la riga i-esima a tutte le altre righe riusciamo ad annullare l' elemento con indici $(j,i)$ della riga j-esima.

$$ \begin{Bmatrix} \frac{f_1}{k_{1,i}} \\\\ \vdots \\\\ \frac{f_{i-1}}{k_{i-1,i}} \\\\ 0 \\\\ \frac{f_{i+1}}{k_{i+1,i}} \\\\ \vdots \\\\ \frac{f_n}{k_{n,i}} \end{Bmatrix} = \begin{bmatrix} \frac{k_{1,1}}{k_{1,i}} - \frac{k_{i,1}}{k_{i,i}} & \dots & \frac{k_{1,i-1}}{k_{1,i}} - \frac{k_{i,i-1}}{k_{i,i}} & 0 & \frac{k_{1,i+1}}{k_{1,i}} - \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{1,n}}{k_{1,i}} - \frac{k_{i,n}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ \frac{k_{i-1,1}}{k_{i-1,i}} - \frac{k_{i,1}}{k_{i,i}}& \dots & \frac{k_{i-1,i-1}}{k_{i-1,i}} - \frac{k_{i,i-1}}{k_{i,i}} & 0 & \frac{k_{i-1,i+1}}{k_{i-1,i}} - \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{i-1,n}}{k_{i-1,i}} - \frac{k_{i,n}}{k_{i,i}}\\\\ \frac{k_{i,1}}{k_{i,i}} & \dots & \frac{k_{i,i-1}}{k_{i,i}} & 1 & \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{i,n}}{k_{i,i}}\\\\ \frac{k_{i+1,1}}{k_{i+1,i}} - \frac{k_{i,1}}{k_{i,i}}& \dots & \frac{k_{i+1,i-1}}{k_{i+1,i}} - \frac{k_{i,i-1}}{k_{i,i}}& 0 & \frac{k_{i+1,i+1}}{k_{i+1,i}} - \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{i+1,n}}{k_{i+1,i}} - \frac{k_{i,n}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ \frac{k_{n,1}}{k_{n,i}} - \frac{k_{i,1}}{k_{i,i}}& \dots & \frac{k_{n,i-1}}{k_{n,i}} - \frac{k_{i,i-1}}{k_{i,i}} & 0 & \frac{k_{n,i+1}}{k_{n,i}} - \frac{k_{i,i+1}}{k_{i,i}}& \dots & \frac{k_{n,n}}{k_{n,i}} - \frac{k_{i,n}}{k_{i,i}} \end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1}\\\\ \eta_{i} \\\\ \eta_{i+1}\\\\ \vdots \\\\ \eta_n \end{Bmatrix} + \begin{Bmatrix} \frac{f_{0,1}}{k_{1,i}} - \frac{f_{0,i}}{k_{i,i}} \\\\ \vdots \\\\ \frac{f_{0,i-1}}{k_{i-1,i}} - \frac{f_{0,i}}{k_{i,i}}\\\\ \frac{f_{0,i}}{k_{i,i}} \\\\ \frac{f_{0,i+1}}{k_{i+1,i}} - \frac{f_{0,i}}{k_{i,i}}\\\\ \vdots \\\\ \frac{f_{0,n}}{k_{n,i}} - \frac{f_{0,i}}{k_{i,i}} \end{Bmatrix}$$

$$ \begin{Bmatrix} f_1 \\\\ \vdots \\\\ f_{i-1} \\\\ 0 \\\\ f_{i+1} \\\\ \vdots \\\\ f_{n} \end{Bmatrix} = \begin{bmatrix} k_{1,1} - \frac{k_{i,1} \, k_{1,i}}{k_{i,i}} & \dots & k_{1,i-1} - \frac{k_{i,i-1} \, k_{1,i}}{k_{i,i}} & 0 & k_{1,i+1} - \frac{k_{i,i+1} \, k_{1,i}}{k_{i,i}} & \dots & k_{1,n} - \frac{k_{i,n} \, k_{1,i}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{i-1,1} - \frac{k_{i,1} \, k_{i-1,i}}{k_{i,i}}& \dots & k_{i-1,i-1} - \frac{k_{i,i-1} \, k_{i-1,i}}{k_{i,i}} & 0 & k_{i-1,i+1} - \frac{k_{i,i+1} \, k_{i-1,i}}{k_{i,i}} & \dots & k_{i-1,n} - \frac{k_{i,n} \, k_{i-1,i}}{k_{i,i}}\\\\ \frac{k_{i,1}}{k_{i,i}} & \dots & \frac{k_{i,i-1}}{k_{i,i}} & 1 & \frac{k_{i,i+1}}{k_{i,i}} & \dots & \frac{k_{i,n}}{k_{i,i}}\\\\ k_{i+1,1} - \frac{k_{i,1} \, k_{i+1,i}}{k_{i,i}}& \dots & k_{i+1,i-1} - \frac{k_{i,i-1} \, k_{i+1,i}}{k_{i,i}}& 0 & k_{i+1,i+1} - \frac{k_{i,i+1} \, k_{i+1,i}}{k_{i,i}} & \dots & k_{i+1,n} - \frac{k_{i,n} \, k_{i+1,i}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{n,1} - \frac{k_{i,1} \, k_{n,i}}{k_{i,i}}& \dots & k_{n,i-1} - \frac{k_{i,i-1} \, k_{n,i}}{k_{i,i}} & 0 & k_{n,i+1} - \frac{k_{i,i+1} \, k_{n,i}}{k_{i,i}}& \dots & k_{n,n} - \frac{k_{i,n} \, k_{n,i}}{k_{i,i}} \end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1}\\\\ \eta_{i} \\\\ \eta_{i+1}\\\\ \vdots \\\\ \eta_n \end{Bmatrix} + \begin{Bmatrix} f_{0,1} - \frac{f_{0,i} \, k_{1,i}}{k_{i,i}} \\\\ \vdots \\\\ f_{0,i-1} - \frac{f_{0,i} \, k_{i-1,i}}{k_{i,i}}\\\\ \frac{f_{0,i}}{k_{i,i}} \\\\ f_{0,i+1} - \frac{f_{0,i} \, k_{i+1,i}}{k_{i,i}}\\\\ \vdots \\\\ f_{0,n} - \frac{f_{0,i} \, k_{n,i}}{k_{i,i}} \end{Bmatrix}$$

Dopo aver eseguito i suddetti passaggi su tutti i gradi di libertà svincolati, per comodità portiamo a primo membro gli spostamenti dei gradi di libertà svincolati.

$$ \begin{Bmatrix} f_1 \\\\ \vdots \\\\ f_{i-1} \\\\ \eta_{i} \\\\ f_{i+1} \\\\ \vdots \\\\ f_{n} \end{Bmatrix} = \begin{bmatrix} k_{1,1} - \frac{k_{i,1} \, k_{1,i}}{k_{i,i}} & \dots & k_{1,i-1} - \frac{k_{i,i-1} \, k_{1,i}}{k_{i,i}} & 0 & k_{1,i+1} - \frac{k_{i,i+1} \, k_{1,i}}{k_{i,i}} & \dots & k_{1,n} - \frac{k_{i,n} \, k_{1,i}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{i-1,1} - \frac{k_{i,1} \, k_{i-1,i}}{k_{i,i}}& \dots & k_{i-1,i-1} - \frac{k_{i,i-1} \, k_{i-1,i}}{k_{i,i}} & 0 & k_{i-1,i+1} - \frac{k_{i,i+1} \, k_{i-1,i}}{k_{i,i}} & \dots & k_{i-1,n} - \frac{k_{i,n} \, k_{i-1,i}}{k_{i,i}}\\\\ - \frac{k_{i,1}}{k_{i,i}} & \dots & - \frac{k_{i,i-1}}{k_{i,i}} & 1 & - \frac{k_{i,i+1}}{k_{i,i}} & \dots & - \frac{k_{i,n}}{k_{i,i}}\\\\ k_{i+1,1} - \frac{k_{i,1} \, k_{i+1,i}}{k_{i,i}}& \dots & k_{i+1,i-1} - \frac{k_{i,i-1} \, k_{i+1,i}}{k_{i,i}}& 0 & k_{i+1,i+1} - \frac{k_{i,i+1} \, k_{i+1,i}}{k_{i,i}} & \dots & k_{i+1,n} - \frac{k_{i,n} \, k_{i+1,i}}{k_{i,i}}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\\\ k_{n,1} - \frac{k_{i,1} \, k_{n,i}}{k_{i,i}}& \dots & k_{n,i-1} - \frac{k_{i,i-1} \, k_{n,i}}{k_{i,i}} & 0 & k_{n,i+1} - \frac{k_{i,i+1} \, k_{n,i}}{k_{i,i}}& \dots & k_{n,n} - \frac{k_{i,n} \, k_{n,i}}{k_{i,i}} \end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1}\\\\ 0 \\\\ \eta_{i+1}\\\\ \vdots \\\\ \eta_n \end{Bmatrix} + \begin{Bmatrix} f_{0,1} - \frac{f_{0,i} \, k_{1,i}}{k_{i,i}} \\\\ \vdots \\\\ f_{0,i-1} - \frac{f_{0,i} \, k_{i-1,i}}{k_{i,i}}\\\\ - \frac{f_{0,i}}{k_{i,i}} \\\\ f_{0,i+1} - \frac{f_{0,i} \, k_{i+1,i}}{k_{i,i}}\\\\ \vdots \\\\ f_{0,n} - \frac{f_{0,i} \, k_{n,i}}{k_{i,i}} \end{Bmatrix}$$


scienza_costruzioni/fem/vincoli_interni.txt · Ultima modifica: 2012/12/02 19:15 (modifica esterna)

Facebook Twitter Google+ Digg Reddit LinkedIn StumbleUpon Email