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

Considérons un trépied composé d’une poutre spatiale tétraédrique avec une rotule à rotule en C.
Si l’on donne la hauteur et la longueur des éléments horizontaux et verticaux, quelle force agit sur les éléments BC et BA, en considérant qu’une force de traction est appliquée en D ?
Un schéma à corps libre est dessiné, comprenant toutes les forces de réaction en A, B et C.
La condition d’équilibre du moment à l’articulation C est appliquée. Les distances sont exprimées en termes de vecteurs de position en trois dimensions.
En simplifiant davantage et en utilisant les conditions d’équilibre des forces, les composantes vectorielles le long de i donnent FAy. De même, en égalisant les coefficients k à zéro, on obtient FBx. 
Enfin, en égalisant les coefficients j, on donne FAx. 
Maintenant, on considère qu’un diagramme à corps libre à l’articulation B permet de calculer les forces FBC et FBA.
Les forces FBD, FBC et FBA peuvent être exprimées à l’aide des vecteurs position.
En appliquant la condition d’équilibre des forces à l’articulation B et en égalisant les coefficients des vecteurs unitaires i, j et k à zéro, on obtient les forces le long de BC et BA.

<ol>
	<li>Hibbeler, R.C. (2016). Génie mécanique \u2012 Statique et dynamique. Hoboken, New Jersey : Pearson Prentice Hall. p. 301.</li>
	<li>Bière, F.P. ; Johnston, E.R. ; Mazurek, D.F ; Cromwell, P.J. et Self, B.P. (2019). Mécanique vectorielle pour les ingénieurs \u2012 Statique et dynamique. New York : McGraw-Hill. p. 306</li>
	<li>Meriam, J.L., Kraige, L.G. et Bolton, J.N. (2020). Génie mécanique \u2012 Statique. Hoboken, New Jersey : John Wiley. p. 193</li>
</ol>

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poutres spatiales : résolution de problèmes

Une poutre spatiale est l'équivalent tridimensionnel d'une poutre plane. Ces structures sont constituées de membres connectés à leurs extrémités, utilisant souvent des articulations à rotule pour créer un cadre stable et polyvalent. En raison de son adaptabilité et de sa capacité à supporter des charges complexes, la poutre spatiale est largement utilisée dans différents projets de construction.

Considérons un trépied constitué d'une poutre spatiale tétraédrique avec une articulation à rotule au point C. Supposons que la hauteur et les longueurs des membres horizontaux et verticaux soient connues.

<img alt="Figure 1" src="/files/ftp_upload/14289/14289_PT_Fig_1v2.svg" style="text-align: center; height: 350px;">

On suppose qu'une force de traction est appliquée au point D. Un diagramme de corps libre qui inclut toutes les forces de réaction au niveau des articulations A, B et C peut être dessiné pour déterminer la force agissant sur les membres BC et BA.

La condition d'équilibre des moments au niveau de l'articulation C est appliquée, en tenant compte des distances exprimées par des vecteurs de position en trois dimensions.

<img alt="Equation 1" src="/files/ftp_upload/14289/14289_PT_Equation_1v3.svg" style="text-align: center; height: 15px;">

En simplifiant davantage et en utilisant les conditions d'équilibre des forces, les composantes vectorielles le long de i fournissent FAy comme étant de 6 N et le long de k donne FBx comme étant de -7,2 N. Enfin, en égalant les coefficients j, on obtient la valeur de FAx comme étant de 6 N.

Maintenant, considérons le diagramme de corps libre au niveau de l'articulation B pour calculer les forces BC et BA. Les forces BD, BC et BA peuvent être exprimées à l'aide de vecteurs de position. La condition d'équilibre des forces au niveau de l'articulation B est appliquée.

<img alt="Equation 2" src="/files/ftp_upload/14289/14289_PT_Equation_2v2.svg" style="text-align: center; height: 13px;">

En égalant les coefficients des vecteurs unitaires i, j et k à zéro, on trouve que les forces le long de BC et BA sont nulles.

Poutres spatiales : Résolution de problèmes

Une poutre spatiale est une contrepartie tridimensionnelle d’une poutre plane. Ces structures se composent de membres reliés à leurs extrémités, utilisant souvent ...

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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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machines

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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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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3D Truss Force Analysis with Joint Equilibrium

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