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Example 1

Given is the solution vector $ \ mathbf {Q} $ for a beam divided into three finite elements as in the diagram below. Using FEM calculate the deflection value for $ X-9 \ mathrm {~ m} $. $ \ mathbf {Q} = \ left \ {\ begin {array} {llllllll} 0 & 0 & 2,5 & 2 & 0 & 6 & 0 & 0 \ end {array} \ right \} \ cdot 10 ^ { 2} $

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Solution

\begin{aligned} &Q:=\left[\begin{array}{c} 0 \\ 0 \\ 2.5 \\ 2 \\ 0 \\ 6 \\ 0 \\ 0 \end{array}\right] \cdot 10^{-2} \\ &L_{1}:=3 \quad m \\ &L_{2}:=4 \quad m \\ &L_{3}:=3 \quad m \\ &X:=9 \quad m \\ &\xi:=\frac{X-L_{1}-L_{2}}{L_{3}} \rightarrow \frac{2}{3} \\ &N_{1}:=1-3 \cdot \xi^{2}+2 \cdot \xi^{3}=0.259 \\ &N_{2}:=L_{3} \cdot\left(\xi-2 \cdot \xi^{2}+\xi^{3}\right)=0.222 \\ &N_{3}:=3 \cdot \xi^{2}-2 \cdot \xi^{3}=0.741 \\ &N_{4}:=L_{3} \cdot\left(\xi^{3}-\xi^{2}\right)=-0.444 \\ &v:=N_{1} \cdot q_{0}+N_{2} \cdot q_{1}+N_{3} \cdot q_{2}+N_{4} \cdot q_{3}=0.01333333 m \end{aligned}