Name: 3d truss force analysis with joint equilibrium
Uploaded: 2023-09-22T11:49:24.000Z
Duration: 1 min 29 s
Channel: JoVE

Stellen Sie sich ein Stativ vor, das aus einem tetraedrischen Weltraumbinder mit einem Kugelgelenk bei C besteht.
Wenn die Höhe und die Längen der horizontalen und vertikalen Stäbe angegeben sind, welche Kraft wirkt auf die Stäbe BC und BA, wenn man bedenkt, dass bei D eine Zugkraft aufgebracht wird?
Es wird ein Freikörperdiagramm gezeichnet, das alle Reaktionskräfte an A, B und C enthält.
Es wird die Momentengleichgewichtsbedingung am Gelenk C angewendet. Die Entfernungen werden in Form von Positionsvektoren in drei Dimensionen ausgedrückt.
Vereinfacht und unter Verwendung der Kraftgleichgewichtsbedingungen ergeben die Vektorkomponenten entlang iF Ay. In ähnlicher Weise ergibt die Gleichsetzung der k Koeffizienten mit Null FBx. 
Wenn man schließlich die j-Koeffizienten gleichsetzt, erhält man FAx. 
Nun wird ein Freikörperdiagramm am Gelenk B betrachtet, um die Kräfte FBC und FBA zu berechnen.
Die Kräfte FBD, FBC und FBA können mit Hilfe der Ortsvektoren ausgedrückt werden.
Wendet man die Kraftgleichgewichtsbedingung an Gelenk B an und setzt man die Koeffizienten der Einheitsvektoren i, j und k mit Null gleich, erhält man die Kräfte entlang von BC und BA.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seite 301.</li>
	<li>Bier, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, P.J. und Self, B.P. (2019). Vektormechanik für Ingenieure \u2012 Statik und Dynamik. New York: McGraw-Hill. Seite 306</li>
	<li>Meriam, J.L., Kraige, L.G. und Bolton, J.N. (2020). Technische Mechanik \u2012 Statik. Hoboken, New Jersey: John Wiley. Seite 193</li>
</ol>

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space trägersysteme: problem lösung

Ein Raumträgersystem ist das dreidimensionale Gegenstück eines ebenen Trägersystems. Diese Strukturen bestehen aus Bauteilen, die an ihren Enden verbunden sind und häufig Kugelgelenke verwenden, um ein stabiles und vielseitiges Rahmenwerk zu schaffen. Aufgrund seiner Anpassungsfähigkeit und seiner Fähigkeit, komplexe Lasten zu tragen, wird das Raumträgersystem in verschiedenen Bauprojekten weit verbreitet eingesetzt. 
 Betrachten Sie ein Stativ, das aus einem tetraedrischen Raumträgersystem mit einem Kugelgelenk am Punkt C besteht. Nehmen wir an, die Höhe und die Längen der horizontalen und vertikalen Bauteile sind bekannt. <img alt="Figure 1" src="/files/ftp_upload/14289/14289_PT_Fig_1v2.svg" style="text-align: center; height: 350px;"> Unter der Annahme, dass eine Zugkraft am Gelenk D angewendet wird, kann eine Freikörperdarstellung erstellt werden, die alle Reaktionskräfte an den Gelenken A, B, und C enthält, um die auf die Bauteile BC und BA wirkenden Kräfte zu bestimmen. <img alt="Equation 1" src="/files/ftp_upload/14289/14289_PT_Equation_1v3.svg" style="text-align: center; height: 15px;"> Die Momentengleichgewichtsbedingung am Gelenk C wird unter Berücksichtigung der Positionsrvektoren in drei Dimensionen angewendet. Weitere Vereinfachungen und die Verwendung der Kräftegleichgewichtsbedingungen ergeben die Vektoranteile entlang der i-Achse als 6 N und entlang der k-Achse als -7,2 N. Durch das Gleichsetzen der j-Koeffizienten wird der Wert der Achse j als 6 N ermittelt. Nun betrachten wir das Freikörperdiagramm am Gelenk B, um die Kräfte BC und BA zu berechnen. Die Kräfte BD, BC und BA können mithilfe von Positionsrvektoren ausgedrückt werden. Die Kräftegleichgewichtsbedingung am Gelenk B wird angewendet. <img alt="Equation 2" src="/files/ftp_upload/14289/14289_PT_Equation_2v2.svg" style="text-align: center; height: 13px;"> Durch das Gleichsetzen der Koeffizienten der i-, j- und k-Einheitsvektoren mit Null ergeben sich die Kräfte entlang von BC und BA als null.

Space Trägersysteme: Problem Lösung

Ein Weltraumfachwerk ist ein dreidimensionales Gegenstück zu einem planaren Fachwerk. Diese Strukturen bestehen aus Stäben, die an ihren Enden miteinander verbunden sind, oft unter Verwendung von ...

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Space Trusses: Problem Solving

3d truss force analysis with joint equilibrium

A space truss is a three-dimensional truss structure made of members connected at their ends. It often uses ball-and-socket joints to form a stable and versatile framework. Because it can handle complex loads, this type of structure is used in many construction projects.

The example begins with a tripod made from a tetrahedral space truss. A ball-and-socket joint is located at C. The height and the lengths of the horizontal and vertical members are known. A tensile force is applied at joint D. A free-body diagram is then drawn to include the reaction forces at joints A, B, and C.

To find the forces in members BC and BA, moment equilibrium is applied at joint C. The distances are written as position vectors in three dimensions. After the equations are simplified, the i component gives F Ay as 6 N, and the k component gives F Bx as -7.2 N. The j component then gives F Ax as 6 N.

The analysis continues with a free-body diagram at joint B. The forces F BD, F BC, and F BA are written using position vectors. Force equilibrium is applied at joint B. When the i, j, and k unit-vector coefficients are set to zero, the forces in BC and BA are both found to be zero.

3D Truss Force Analysis with Joint Equilibrium

3d-truss-force-analysis-with-joint-equilibrium

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space trusses: problem solving

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key terms and concepts

Analyze a space truss using free-body diagrams, position vectors, and joint equilibrium to find reaction forces and determine member forces in a 3D structure.

3D Truss Force Analysis with Joint Equilibrium | JoVE K12

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3D Truss Force Analysis with Joint Equilibrium

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