Name: finding resultant loads on a beam
Uploaded: 2023-09-22T11:49:22.000Z
Duration: 1 min 29 s
Channel: JoVE

Poutre de longueur L soumise à une charge variable, qui est une combinaison de répartition parabolique et trapézoïdale des charges le long de l’axe des x.
Dans la charge parabolique, considérons un élément différentiel de force dR sur la longueur dx. La charge résultante est obtenue en intégrant l’élément différentiel sur la longueur de la charge. Il est situé au centre de l'aire parabolique, et la coordonnée x du centroïde est déterminée à l'aide du principe du moment.
La zone trapézoïdale peut être divisée en zones rectangulaires et triangulaires, la charge résultante agissant au niveau de leurs centroïdes. Ces centroïdes sont positionnés à la moitié de la longueur du rectangle et à un tiers de la longueur de la base à partir du côté vertical du triangle.
La charge résultante pour la zone trapézoïdale est obtenue en additionnant les charges résultantes individuelles et passe par le centroïde de la surface, déterminé par le principe du moment.
La charge répartie totale est la somme des charges résultantes des aires trapézoïdale et parabolique, passant par le centroïde de l’aire combinée, déterminée par le principe de moment.

<ol>
	<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. 264</li>
	<li>Meriam, J.L. ; Kraige, L.G. et Bolton, J.N. (2020). Génie mécanique \u2012 Statique. Hoboken, New Jersey : John Wiley. p. 273</li>
	<li>Hibbeler, R.C. (2016). Génie mécanique \u2012 Statique et dynamique. Hoboken, New Jersey : Pearson Prentice Hall. pp EM-2</li>
</ol>

chargement le long d'un seul axe

En génie civil, l'analyse des poutres soumises à des charges variables est un aspect crucial pour comprendre le comportement et les performances de ces éléments structuraux. Un scénario courant implique une poutre soumise à une combinaison de différentes distributions de charge. Considérons une poutre de longueur L soumise à une charge variable, qui est une combinaison de distribution de charge parabolique et trapézoïdale le long de l'axe x. Dans ce cas, il est essentiel de déterminer les charges résultantes, leurs emplacements et le centre de gravité de la zone combinée pour prédire la réponse de la poutre sous ces conditions de charge. Tout d'abord, examinons la distribution de charge parabolique. Considérons un élément différentiel de force dR agissant sur une petite longueur dx. Pour déterminer la charge résultante pour la zone parabolique, l'élément différentiel dR est intégré sur toute la longueur de la charge. L'emplacement de cette charge résultante est au centre de gravité de la zone parabolique. Le principe des moments est appliqué pour trouver la coordonnée x du centre de gravité, qui stipule que le premier moment de la zone autour d'un axe est égal au produit de la zone et de la distance de son centre de gravité par rapport à l'axe. Ensuite, la distribution de charge trapézoïdale est analysée en la divisant en deux régions rectangulaires et triangulaires. Les charges résultantes pour ces régions individuelles agissent à leurs centres de gravité respectifs. Pour la zone rectangulaire, le centre de gravité se trouve à la moitié de la longueur du rectangle. Pour la zone triangulaire, le centre de gravité est situé à un tiers de la longueur de la base à partir du côté vertical du triangle. En additionnant les charges résultantes individuelles pour les zones rectangulaires et triangulaires, on peut déterminer la charge résultante pour toute la zone trapézoïdale. L'emplacement de la charge résultante passe par le centre de gravité de la zone trapézoïdale, qui peut également être déterminé en utilisant le principe des moments. De même, en ajoutant les charges résultantes des zones parabolique et trapézoïdale, on peut déterminer la charge résultante totale agissant sur la poutre. L'emplacement de la charge résultante totale passe par le centre de gravité de la zone combinée, qui peut être déterminé plus précisément en utilisant le principe des moments.

Chargement le long d'un seul axe

En ingénierie structurelle, l’analyse des poutres soumises à des charges variables est un aspect essentiel pour comprendre le comportement et les performances de ces ...

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finding resultant loads on a beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted.

The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the full length of the load. The resultant load for the parabolic area acts at the centroid of that area. The x-coordinate of the centroid is found with the moment principle, which says the first moment of an area about an axis equals the area times the distance to its centroid.

The trapezoidal load is then split into two simpler shapes. One part is rectangular, and the other is triangular. Their resultant loads act at their own centroids. For the rectangle, the centroid lies halfway along its length. For the triangle, the centroid is one-third of the base length from the vertical side.

After those parts are found, their loads are added to get the resultant load of the full trapezoidal area. That load passes through the centroid of the trapezoid, which can also be found with the moment principle. Finally, the resultant loads from the parabolic and trapezoidal areas are combined. Their sum gives the total load on the beam, and its location passes through the centroid of the combined area.

Finding Resultant Loads on a Beam

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key terms and concepts

Learn how parabolic and trapezoidal loads on a beam are broken into simpler areas, then combined to find the resultant force and centroid.

Finding Resultant Loads on a Beam | JoVE K12

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Finding Resultant Loads on a Beam

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