B7 a
$$
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\nonumber$$
Lösning
Reaktionskrafter
\[\eqright H_\mathrm{A} = 0\]
Gör ett snitt genom leden och frilägg den vänstra delen. Momentjämvikt ger (med villkoret att \(M(2L)=0\), pga leden)
\[\eqcwmom{x=2L} M(2L) + \R{A}\, 2L=0 \gives \R{A} = 0\]
\[\eqcwmom{C} \R{A}\,6L + \R{B}\,2L - P\, 4L = 0 \gives \R{B} = 2P\]
\[\equp \R{A} + \R{B} + \R{C} - P = 0 \gives \R{CB} = P - \R{B} = -P\]
Notera: Man behöver inte snitta vid leden och friläga vänstra eller högra sidan, det går lika bra att ställa upp momentjämvikt kring leden för hela strukturen.
Snittkrafter
Tre snitt behövs, ett på vardera sidan om punktlasten samt ett mellan stöd B och C.
Snitt 1
\(0 \le x \le 2L\)
\[\equp T(x) = -\R{A} =0 \]
\[\eqcwmom{x} M(x) = -\R{A} \, x = 0 \]
Snitt 2
\(2L \le x \le 4L\)
\[\equp T(x) = P \]
\[\eqcwmom{x} M(x) = P \, (x-2L) \]
Snitt 3
\(4L \le x \le 6L\)
\[\equp T(x) = P - \R{B} = P -2P = -P \]
\[\begin{align}
\eqcwmom{x} M(x) &= P \, (x-2L) - \R{B} \, (x-4L) \nonumber \newline &= P \, (x-2L) - 2P\, (x-4L) = -P (x - 6L) \end{align}\]
Skissa diagrammet genom att evaluera uttrycken ovan för några punkter, lämpligtvis i övergångarna mellan snitten samt vid ändpunkterna. Kontrollera att momentdiagrammet är kontinuerligt.