Aufgabenstellung Wir untersuchen die Belastung eines ebenen Stabwerks. Die Stäbe haben wie skizziert die Länge ℓ bzw. ℓ/2.
Die Struktur wird mit der Kraft F belastet.
Caption Gesucht ist ein Vergleich zwischen der klassischen Stabwerkstheorie und einer Herangehensweise, bei der wir eine feste Verbindung der Stäbe in den Knoten ansetzten. Grundlage des Modells ist die FEM-Lösung der Felddifferentialgleichung im Vergleich zur Lösung in Problemstellung „Stab“.
Wir stellen das Modell des Stabwerks mit dem Prinzip der virtuellen Verrückungen auf und vergleichen, wie sich diese von der Herangehensweise aus „Stab“ mit der analytischen Lösung unterscheidet.
Lösung mit Maxima Wir nutzen das Computer-Algebra-System Maxima zur Lösung. Das macht hier Sinn, weil wir die Herangehensweise mit der aus Stab vergleichen wollen – für die wir ebenfalls Maxima eingesetzt haben.
Declarations Wir übernehmen alle Vereinbarungen und Parameter aus der Problemformulierung „Stab“.
Gleichgewichtsbedingungen Für die Gleichgewichtsbedingung nach dem
Delta W = 0
= delta \Pi - \delta Wa
benötigen wir die virtuelle Formänderungsenergie \delta \Pi und die virtuelle Arbeit der äußeren Kraft F.
Mit allen Konventionen für die Knoten-Verschiebungen ist
Delta Wa = -delta W_{4,0} *F
Für \delta \Pi gilt
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0
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3
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⋅
(
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1
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1
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=
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{\displaystyle {\begin{pmatrix}{\frac {2{{A}^{2}}E\eta }{3{\ell _{0}}}}&0&-\left({\frac {2{{A}^{2}}E\eta }{{\ell }_{0}^{2}}}\right)&0&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&0&0&0\\0&{\frac {2{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{2{{\ell }_{0}^{2}}}}\right)&0&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}\\-\left({\frac {2{{A}^{2}}E\eta }{{\ell }_{0}^{2}}}\right)&-\left({\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta +3{{\ell }_{0}^{2}}AE}{2{{\ell }_{0}^{3}}}}+{\frac {8{{A}^{2}}E\eta }{{\ell }_{0}^{3}}}&0&-\left({\frac {2{{A}^{2}}E\eta }{{\ell }_{0}^{2}}}\right)&-\left({\frac {{{A}^{2}}E\eta +3{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}&{\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\\0&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&0&{\frac {3{{A}^{2}}E\eta +{{\ell }_{0}^{2}}AE}{2{{\ell }_{0}^{3}}}}+{\frac {2AE}{\ell _{0}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{2{{\ell }_{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}&-\left({\frac {3{{A}^{2}}E\eta +{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)\\{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {2{{A}^{2}}E\eta }{{\ell }_{0}^{2}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{2{{\ell }_{0}^{2}}}}\right)&{\frac {4{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}\\0&-\left({\frac {{{A}^{2}}E\eta }{2{{\ell }_{0}^{2}}}}\right)&-\left({\frac {{{A}^{2}}E\eta +3{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta +3{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}+{\frac {{{A}^{2}}E\eta }{{\ell }_{0}^{3}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&-\left({\frac {3{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)\\0&0&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}&-\left({\frac {3{{A}^{2}}E\eta +{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}\right)&{\frac {3{{A}^{2}}E\eta +{{\ell }_{0}^{2}}AE}{4{{\ell }_{0}^{3}}}}+{\frac {AE}{\ell _{0}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\\0&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&{\frac {{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {3{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{{\ell }_{0}^{2}}}}&{\frac {2{{A}^{2}}E\eta }{3{\ell _{0}}}}\end{pmatrix}}\cdot {\begin{pmatrix}{W_{1,0}}\\{U_{1,0}}\\{{\Phi }_{1,0}}\\{W_{2,0}}\\{U_{2,0}}\\{{\Phi }_{2,0}}\\{W_{3,0}}\\{U_{3,0}}\\{{\Phi }_{3,0}}\\{W_{4,0}}\\{U_{4,0}}\\{{\Phi }_{4,0}}\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\\0\\F\\0\\0\end{pmatrix}}}
Hier kommt jetzt irgendein Text.
S
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{\displaystyle SomeText}
Title Text
Element-Steigigkeitsmatrizen mit globalen Koordinaten Element #1
k
1
=
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{\displaystyle {k_{1}}={\begin{pmatrix}{\frac {8{{A}^{2}}E\eta }{\ell _{0}^{3}}}&0&{\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}&-\left({\frac {8{{A}^{2}}E\eta }{\ell _{0}^{3}}}\right)&0&{\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}\\0&{\frac {2AE}{\ell _{0}}}&0&0&-\left({\frac {2AE}{\ell _{0}}}\right)&0\\{\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}&0&{\frac {2{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}\right)&0&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}\\-\left({\frac {8{{A}^{2}}E\eta }{\ell _{0}^{3}}}\right)&0&-\left({\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}\right)&{\frac {8{{A}^{2}}E\eta }{\ell _{0}^{3}}}&0&-\left({\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}\right)\\0&-\left({\frac {2AE}{\ell _{0}}}\right)&0&0&{\frac {2AE}{\ell _{0}}}&0\\{\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}&0&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {2{{A}^{2}}E\eta }{\ell _{0}^{2}}}\right)&0&{\frac {2{{A}^{2}}E\eta }{3{\ell _{0}}}}\end{pmatrix}}}
Element #2
k
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=
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)
{\displaystyle {k_{2}}={\begin{pmatrix}{\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\\{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\\{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}\\-\left({\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)\\-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)\\{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}\end{pmatrix}}}
Element #3
k
3
=
(
A
2
E
η
+
3
ℓ
0
2
A
E
4
ℓ
0
3
−
(
3
A
2
E
η
−
3
ℓ
0
2
A
E
4
ℓ
0
3
)
A
2
E
η
4
ℓ
0
2
−
(
A
2
E
η
+
3
ℓ
0
2
A
E
4
ℓ
0
3
)
3
A
2
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η
−
3
ℓ
0
2
A
E
4
ℓ
0
3
A
2
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η
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ℓ
0
2
−
(
3
A
2
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η
−
3
ℓ
0
2
A
E
4
ℓ
0
3
)
3
A
2
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η
+
ℓ
0
2
A
E
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ℓ
0
3
−
(
3
A
2
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η
4
ℓ
0
2
)
3
A
2
E
η
−
3
ℓ
0
2
A
E
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{\displaystyle {k_{3}}={\begin{pmatrix}{\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\\-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)\\{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}\\-\left({\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta +3{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)\\{\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&-\left({\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta -{\sqrt {3}}{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}\right)&{\frac {3{{A}^{2}}E\eta +{\ell _{0}^{2}}AE}{4{\ell _{0}^{3}}}}&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\\{\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&-\left({\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}\right)&{\frac {{\sqrt {3}}{{A}^{2}}E\eta }{4{\ell _{0}^{2}}}}&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}\end{pmatrix}}}
Element #4
k
4
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0
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{\displaystyle {k_{4}}={\begin{pmatrix}{\frac {{{A}^{2}}E\eta }{\ell _{0}^{3}}}&0&{\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}&-\left({\frac {{{A}^{2}}E\eta }{\ell _{0}^{3}}}\right)&0&{\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}\\0&{\frac {AE}{\ell _{0}}}&0&0&-\left({\frac {AE}{\ell _{0}}}\right)&0\\{\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}&0&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}\right)&0&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}\\-\left({\frac {{{A}^{2}}E\eta }{\ell _{0}^{3}}}\right)&0&-\left({\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}\right)&{\frac {{{A}^{2}}E\eta }{\ell _{0}^{3}}}&0&-\left({\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}\right)\\0&-\left({\frac {AE}{\ell _{0}}}\right)&0&0&{\frac {AE}{\ell _{0}}}&0\\{\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}&0&{\frac {{{A}^{2}}E\eta }{6{\ell _{0}}}}&-\left({\frac {{{A}^{2}}E\eta }{2{\ell _{0}^{2}}}}\right)&0&{\frac {{{A}^{2}}E\eta }{3{\ell _{0}}}}\end{pmatrix}}}
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