Przykład 1

Dla danych jak na rysunku stosując procedurę MES policzyć wektor przemieszczeń w punkcie \(\mathrm{P}(0,1)\). \begin{aligned} & N_1(x, y)=\frac{1}{2}-\frac{x}{4}-\frac{y}{4} \\ & N_2(x, y)=\frac{1}{4}+\frac{x}{4} \\ & N_3(x, y)=\frac{1}{4}+\frac{y}{4} \\ & \mathrm{Q}^e=\{1,2,-0.5,-1,0.2,-0.4\} \cdot 10^{-4} \end{aligned}

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Rozwiązanie

\begin{aligned} & d = \left[\begin{array}{c} 1 \\ 2 \\ -0.5 \\ -1 \\ 0.2 \\ -0.4 \end{array}\right] \cdot 10^{-4} \\ & N(x, y) = \left[\begin{array}{cccccc} N_1(x, y) & 0 & N_2(x, y) & 0 & N_3(x, y) & 0 \\ 0 & N_1(x, y) & 0 & N_2(x, y) & 0 & N_3(x, y) \end{array}\right] \rightarrow \left[\begin{array}{cccccc} -\frac{y}{4}+\left(\frac{1}{2}-\frac{x}{4}\right) & 0 & \frac{x}{4}+\frac{1}{4} & 0 & \frac{y}{4}+\frac{1}{4} & 0 \\ 0 & -\frac{y}{4}+\left(\frac{1}{2}-\frac{x}{4}\right) & 0 & \frac{x}{4}+\frac{1}{4} & 0 & \frac{y}{4}+\frac{1}{4} \end{array}\right] \\ & N(0,1) = \left[\begin{array}{llllll} 0.25 & 0 & 0.25 & 0 & 0.5 & 0 \\ 0 & 0.25 & 0 & 0.25 & 0 & 0.5 \end{array}\right] \\ & u = N(0,1) \cdot d = \left[\begin{array}{l} 2.25 \cdot 10^{-5} \\ 5 \cdot 10^{-6} \end{array}\right] \end{aligned}