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Nedböjning av en konsolbalk med stöd

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Beskrivning

En balk är belastad med en konstant utbredd last med magnitud \(q\). Tvärsnittet är kvadratiskt med sidlängd \(a\).

  • Utgå från balkens differentialekvation och härled utböjningsfunktionen \(w(x)\).
  • Ta fram koordinaten \(\bar{x}\) där \(T(\bar{x})=0\)

Facit

Nedböjning

\[ w(x) = \frac{qL^4}{24EI} \paren{ -\paren{\frac{x}{L}}^4 + \frac{5}{2}\paren{\frac{x}{L}}^3 - \frac{3}{2}\paren{\frac{x}{L}}^2 } \]
  • Tvärkraften är noll för $$ x = \frac{5L}{8} $$

Lösning

Balkens differentialekvation

Integrera balkens differentialekvation fyra gånger $$ \begin{align} w^{IV}(x) &= -\frac{q}{EI} \qgives \\ w^\tris(x) &= -\frac{q}{EI} \ x + C_1 \qgives \\ w^\bis(x) &= -\frac{q}{EI} \ \frac{x^2}{2} + C_1 \ x + C_2 \qgives \\ w'(x) &= -\frac{q}{EI} \ \frac{x^3}{6} + C_1 \frac{x^2}{2} + C_2 x + C_3 \qgives \\ w(x) &= -\frac{q}{EI} \ \frac{x^4}{24} + C_1 \frac{x^3}{6} + C_2 \frac{x^2}{2} + C_3 x + C_4 \end{align} $$

Randvillkor

  • \(w(0) = 0\) \(\Rightarrow\) \(C_4 = 0\)
  • \(w'(0) = 0\) \(\Rightarrow\) \(C_3 = 0\)
  • \(M=-EIw''(L) = 0\) \(\Rightarrow\) \(w''(L) = 0\) \(\Rightarrow\) $$ w^\bis (L) = -\frac{q}{EI} \ \frac{L^2}{2} + C_1 \cdot L + C_2 = 0 \equivalent C_2 = \frac{q}{EI} \ \frac{L^2}{2} - C_1 \cdot L $$
  • \(w(L) = 0\) \(\Rightarrow\) $$ w(L) = -\frac{q}{EI} \ \frac{L^4}{24} + C_1 \frac{L^3}{6} + \paren{ \frac{q}{EI} \ \frac{L^2}{2} - C_1 \cdot L } \frac{L^2}{2} = 0 \gives $$
\[ C_1 = \frac{5qL}{8EI} \gives \]
\[ C_2 = \frac{q}{EI} \ \frac{L^2}{2} - \frac{5qL}{8EI} \cdot L = - \frac{qL^2}{8EI} \]

Den totala nedböjningen fås därmed som $$ \begin{align} w(x) &= -\frac{q}{EI} \ \frac{x^4}{24} + \frac{5qL}{8EI} \frac{x^3}{6} -\frac{qL^2}{8EI} \frac{x^2}{2} \newline &= \frac{qL^4}{24EI} \paren{ -\paren{\frac{x}{L}}^4 + \frac{5}{2}\paren{\frac{x}{L}}^3 - \frac{3}{2}\paren{\frac{x}{L}}^2 } \end{align} $$

Tvärkraft

Tvärkraften fås nu som $$ T(x) = -EI w^\tris (x) = q \ x - \frac{5qL}{8} $$

\[ T(x) = 0 \gives q \ x - \frac{5qL}{8} = 0 \equivalent x = \frac{5L}{8} \]
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