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Example 1

For the analysis of a beam with a length of $ L = 5 \ mathrm {~ m} $, stiffness $ E J = 40,000 \ mathrm {kNm} ^ {2} $ and a homogeneous load $ p = 12 \ mathrm {kN} / \ mathrm {m} $ the Finite Difference Method has been applied. The distance between $$ \ mathbf {v} = \ left [\ begin {array} {r} 0.0012375 \\ 0.0 \\ 0.0012375 \\ 0.0036 \\ 0.0060375 \\ 0.0078 \\ 0.0084375 \\ 0.0078 \\ 0.0060375 \ end {array} \ right] \ mathrm {m} $$ Using the appropriate differential formula, calculate the value of the bending moment $ M $ at the node where $ x = 2 \ mathrm {~ m} $?

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Solution

\begin{aligned} &L=5 \quad m \\ &EI=40000 \quad k N m^{2} \\ &p=12 \quad \frac{k N}{m} \\ &h=1 \quad m \\ &v_{1}=0.0012375 \quad m \\ &v_{2}=0.0036 \quad m \\ &v_{3}=0.0060375 \quad m \\ &M=-E I \cdot \frac{v_{1}-2 \cdot v_{2}+v_{3}}{h^{2}}=-3 \quad k N m \end{aligned}