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qstruct:teoria:qfem:fem:connessioni_elemento-vertice

Connessioni elemento-vertice

Per gestire le varie tipologie di collegamento interno, introduciamo il concetto di connessione elemento-vertice che a sua volta può essere definita nel sistema di riferimento locale o globale. Supponiamo in prima battuta che sia definito nel sistema locale.

Analizziamo le connessioni che un elemento ha con il suo vertice $i$.

Se c'è la connessione, i termini corrispondenti della matrice $\mathbf{K}$ e del vettore $\mathbf{f}_0$ rimangono inalterati e per conoscere il valore della forza nodale $f_i$ dovremo aspettare di avere calcolato tutti gli spostamenti dei vertici.

Se invece la connessione non esiste, sappiamo già che la corrispondente forza nodale $f_i$ è nulla. Di conseguenza possiamo scrivere

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

A questo punto possiamo dividere ciascuna riga $j$ per il corrispondente termine $k_{j,i}$,

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

Se a ciascuna riga $j \ne i$ sottraiamo la riga $i$ la matrice diventa

$$ \begin{bmatrix} k_{1,1} / k_{1,i} - k_{i,1} / k_{i,i}& \dots & k_{1,i-1} / k_{1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{1,i+1} / k_{1,i} - k_{i,i+1} / k_{i,i}& \dots & k_{1,n} / k_{1,i} - k_{i,n} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{i-1,1} / k_{i-1,i} - k_{i,1} / k_{i,i}& \dots & k_{i-1,i-1} / k_{i-1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{i-1,i+1} / k_{i-1,i} - k_{i,i+1} / k_{i,i} & \dots & k_{i-1,n} / k_{i-1,i} - k_{i,n} / k_{i,i}\\\\ k_{i,1} / k_{i,i}& \dots & k_{i,i-1} / k_{i,i}& 1 & k_{i,i+1} / k_{i,i} & \dots & k_{i,n} / k_{i,i}\\\\ k_{i+1,1} / k_{i+1,i} - k_{i,1} / k_{i,i}& \dots & k_{i+1,i-1} / k_{i+1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{i+1,i+1} / k_{i+1,i} - k_{i,i+1} / k_{i,i}& \dots & k_{i+1,n} / k_{i+1,i} - k_{i,n} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{n,1} / k_{n,i} - k_{i,1} / k_{i,i}& \dots & k_{n,i-1} / k_{n,i} - k_{i,i-1} / k_{i,i}& 0 & k_{n,i+1}/ k_{n,i} -k_{i,i+1} / k_{i,i} & \dots & k_{n,n} / k_{n,i} - k_{i,n} / k_{i,i} \end{bmatrix}$$

e il vettore dei termini noti diventa

$$\begin{Bmatrix} f_{0,1} / k_{1,i} - f_{0,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,i-1} / k_{i-1,i} - f_{0,i} / k_{i,i}\\\\ f_{0,i} / k_{i,i}\\\\ f_{0,i+1} / k_{i+1,i} - f_{0,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,n} / k_{n,i} - f_{0,i} / k_{i,i}\\\\ \end{Bmatrix}$$

A questo punto moltiplichiamo ciascuna riga j per il termine $k_{j,i}$, ottenendo la matrice

$$ \mathbf{K}_{C} = \begin{bmatrix} k_{1,1} - k_{i,1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,i-1} - k_{i,i-1} \cdot k_{1,i} / k_{i,i}& 0 & k_{1,i+1} - k_{i,i+1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,n} - k_{i,n} \cdot k_{1,i} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{i-1,1} - k_{i,1} \cdot k_{i-1,i} / k_{i,i}& \dots & k_{i-1,i-1} - k_{i,i-1} \cdot k_{i-1,i} / k_{i,i}& 0 & k_{i-1,i+1} - k_{i,i+1} \cdot k_{i-1,i} / k_{i,i} & \dots & k_{i-1,n} - k_{i,n} \cdot k_{i-1,i} / k_{i,i}\\\\ k_{i,1}& \dots & k_{i,i-1}& k_{i,i} & k_{i,i+1} & \dots & k_{i,n} \\\\ k_{i+1,1} / k_{i+1,i} - k_{i,1} / k_{i,i}& \dots & k_{i+1,i-1} / k_{i+1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{i+1,i+1} / k_{i+1,i} - k_{i,i+1} / k_{i,i}& \dots & k_{i+1,n} / k_{i+1,i} - k_{i,n} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{n,1} / k_{n,i} - k_{i,1} / k_{i,i}& \dots & k_{n,i-1} / k_{n,i} - k_{i,i-1} / k_{i,i}& 0 & k_{n,i+1}/ k_{n,i} -k_{i,i+1} / k_{i,i} & \dots & k_{n,n} / k_{n,i} - k_{i,n} / k_{i,i} \end{bmatrix}$$

e il vettore

$$\mathbf{f}_{0,C} = \begin{Bmatrix} f_{0,1} - f_{0,i} \cdot k_{1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,i-1} - f_{0,i} \cdot k_{i-1,i} / k_{i,i}\\\\ f_{0,i}\\\\ f_{0,i+1} - f_{0,i} \cdot k_{i+1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,n} - f_{0,i} \cdot k_{n,i}/ k_{i,i}\\\\ \end{Bmatrix}$$

con le quali possiamo scrivere

$$\mathbf{f} = \mathbf{K}_{C} \cdot \boldsymbol{\eta} + \mathbf{f}_{0,C}$$

In fase di assemblaggio della matrice globale, notiamo che la forza $f_{i}$ trasmessa alla struttura è nulla, pertanto assumiamo la relazione

$$\begin{Bmatrix} f_1 \\\\ \vdots \\\\ f_{i-1} \\\\ f_{i} \\\\ f_{i+1} \\\\ \vdots \\\\ f_{n} \\\\ \end{Bmatrix} = \begin{bmatrix}K'_{C}\end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1} \\\\ \eta_{i} \\\\ \eta_{i+1} \\\\ \vdots \\\\ \eta_{n} \\\\ \end{Bmatrix} + \begin{Bmatrix} f'_{0,C}\end{Bmatrix} $$

in cui

$$ \mathbf{K'}_{C} = \begin{bmatrix} k_{1,1} - k_{i,1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,i-1} - k_{i,i-1} \cdot k_{1,i} / k_{i,i}& 0 & k_{1,i+1} - k_{i,i+1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,n} - k_{i,n} \cdot k_{1,i} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{i-1,1} - k_{i,1} \cdot k_{i-1,i} / k_{i,i}& \dots & k_{i-1,i-1} - k_{i,i-1} \cdot k_{i-1,i} / k_{i,i}& 0 & k_{i-1,i+1} - k_{i,i+1} \cdot k_{i-1,i} / k_{i,i} & \dots & k_{i-1,n} - k_{i,n} \cdot k_{i-1,i} / k_{i,i}\\\\ 0& \dots & 0& 0 & 0 & \dots & 0 \\\\ k_{i+1,1} / k_{i+1,i} - k_{i,1} / k_{i,i}& \dots & k_{i+1,i-1} / k_{i+1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{i+1,i+1} / k_{i+1,i} - k_{i,i+1} / k_{i,i}& \dots & k_{i+1,n} / k_{i+1,i} - k_{i,n} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{n,1} / k_{n,i} - k_{i,1} / k_{i,i}& \dots & k_{n,i-1} / k_{n,i} - k_{i,i-1} / k_{i,i}& 0 & k_{n,i+1}/ k_{n,i} -k_{i,i+1} / k_{i,i} & \dots & k_{n,n} / k_{n,i} - k_{i,n} / k_{i,i} \end{bmatrix}$$

e

$$\mathbf{f'}_{0,C} = \begin{Bmatrix} f_{0,1} - f_{0,i} \cdot k_{1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,i-1} - f_{0,i} \cdot k_{i-1,i} / k_{i,i}\\\\ 0\\\\ f_{0,i+1} - f_{0,i} \cdot k_{i+1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,n} - f_{0,i} \cdot k_{n,i}/ k_{i,i}\\\\ \end{Bmatrix}$$

In fase di calcolo delle sollecitazioni nodali della trave, invertiamo la lo spostamento $\eta_i$, incognito perché diverso dal corrispondente grado di libertà del vertice, con la forza nodale $f_i$, nulla, ottenendo

$$\begin{Bmatrix} f_1 \\\\ \vdots \\\\ f_{i-1} \\\\ \eta_{i} \\\\ f_{i+1} \\\\ \vdots \\\\ f_{n} \\\\ \end{Bmatrix} = \begin{bmatrix}K''_{C}\end{bmatrix} \begin{Bmatrix} \eta_1 \\\\ \vdots \\\\ \eta_{i-1} \\\\ f_{i} \\\\ \eta_{i+1} \\\\ \vdots \\\\ \eta_{n} \\\\ \end{Bmatrix} + \begin{Bmatrix} f''_{0,C}\end{Bmatrix} $$

in cui

$$ \mathbf{K''}_{C} = \begin{bmatrix} k_{1,1} - k_{i,1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,i-1} - k_{i,i-1} \cdot k_{1,i} / k_{i,i}& 0 & k_{1,i+1} - k_{i,i+1} \cdot k_{1,i} / k_{i,i}& \dots & k_{1,n} - k_{i,n} \cdot k_{1,i} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{i-1,1} - k_{i,1} \cdot k_{i-1,i} / k_{i,i}& \dots & k_{i-1,i-1} - k_{i,i-1} \cdot k_{i-1,i} / k_{i,i}& 0 & k_{i-1,i+1} - k_{i,i+1} \cdot k_{i-1,i} / k_{i,i} & \dots & k_{i-1,n} - k_{i,n} \cdot k_{i-1,i} / k_{i,i}\\\\ -k_{i,1} / k_{i,i}& \dots & -k_{i,i-1} / k_{i,i}& 1 / k_{i,i}& -k_{i,i+1} / k_{i,i} & \dots & -k_{i,n} / k_{i,i} \\\\ k_{i+1,1} / k_{i+1,i} - k_{i,1} / k_{i,i}& \dots & k_{i+1,i-1} / k_{i+1,i} - k_{i,i-1} / k_{i,i}& 0 & k_{i+1,i+1} / k_{i+1,i} - k_{i,i+1} / k_{i,i}& \dots & k_{i+1,n} / k_{i+1,i} - k_{i,n} / k_{i,i}\\\\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\\\ k_{n,1} / k_{n,i} - k_{i,1} / k_{i,i}& \dots & k_{n,i-1} / k_{n,i} - k_{i,i-1} / k_{i,i}& 0 & k_{n,i+1}/ k_{n,i} -k_{i,i+1} / k_{i,i} & \dots & k_{n,n} / k_{n,i} - k_{i,n} / k_{i,i} \end{bmatrix}$$

$$\mathbf{f''}_{0,C} = \begin{Bmatrix} f_{0,1} - f_{0,i} \cdot k_{1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,i-1} - f_{0,i} \cdot k_{i-1,i} / k_{i,i}\\\\ -f_{0,i} / k_{i,i} \\\\ f_{0,i+1} - f_{0,i} \cdot k_{i+1,i} / k_{i,i}\\\\ \vdots \\\\ f_{0,n} - f_{0,i} \cdot k_{n,i}/ k_{i,i}\\\\ \end{Bmatrix}$$


qstruct/teoria/qfem/fem/connessioni_elemento-vertice.txt · Ultima modifica: 2013/02/08 10:45 da mickele

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