Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

Considérons une poutre subissant une charge répartie, deux charges concentrées et un moment de couple. Établissez la relation entre le cisaillement et la charge répartie.
Tout d’abord, considérez une section élémentaire sur la poutre exempte de toute charge concentrée ou moment de couple et dessinez un schéma de corps libre de la section.
Pour maintenir l’équilibre, la force de cisaillement agissant sur le côté droit de la section doit être incrémentée d’une quantité faible et finie.
La force résultante de la charge répartie agit à une distance fractionnaire de l’extrémité droite de la section.
En utilisant l’équation d’équilibre de la force verticale, on obtient une relation entre le cisaillement et la charge.
Ensuite, en divisant les deux membres de l’équation par Δx et en laissant Δx approcher de zéro, on peut déterminer la pente de la force de cisaillement, qui est égale à l’intensité de la charge distribuée.
Enfin, la réorganisation de l’équation et l’intégration de la charge répartie sur la section élémentaire entre deux points arbitraires conduisent à une relation entre la variation de cisaillement et l’aire sous la courbe de charge.

<ol>
	<li>Hibbeler, R.C. (2016). Génie mécanique \u2012 Statique et dynamique. Hoboken, New Jersey : Pearson Prentice Hall. p. 370 et 371.</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. Pages 391 et 392.</li>
</ol>

relation entre la charge répartie et la force de cisaillement

Comprendre la relation entre la charge répartie et la force de cisaillement dans l'analyse structurelle est crucial pour analyser les poutres soumises à diverses conditions de charge. Considérons le cas d'une poutre subissant une charge répartie, deux charges ponctuelles et un moment couple. <img alt="Figure 1" src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;"> La relation entre la force de cisaillement et la charge répartie pour le cas donné peut être établie en suivant la procédure donnée. Tout d'abord, considérons une section élémentaire de la poutre exempte de toute charge ponctuelle ou moment couple. Nous pouvons dessiner un diagramme des forces pour analyser les forces agissant sur cette section. Le diagramme comprendra la charge répartie agissant le long de la longueur de la poutre, la force de cisaillement V(x) agissant du côté droit de la section, et une force de cisaillement incrémentale dV ajoutée pour maintenir l'équilibre. Pour maintenir l'équilibre dans la direction verticale, la force de cisaillement agissant du côté droit de la section doit être augmentée d'une petite quantité finie, ΔV. La force résultante de la charge répartie, w(x)Δx, agit à une distance fractionnelle de l'extrémité droite de la section. En utilisant l'équation d'équilibre des forces verticales, on peut obtenir la relation suivante entre la force de cisaillement et la charge : <img alt="Equation 1" src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;"> Ensuite, nous divisons les deux côtés de l'équation par Δx et laissons Δx approcher zéro : <img alt="Equation 2" src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;"> Cette équation montre que la pente de la force de cisaillement est égale à l'intensité de la charge répartie. Enfin, nous pouvons réarranger l'équation et intégrer la charge répartie sur la section élémentaire entre deux points arbitraires, Q et R : <img alt="Equation 3" src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;"> Cette équation intégrale montre la relation entre la variation de la force de cisaillement et l'aire sous la courbe de charge. La différence de force de cisaillement entre les points Q et R est égale à l'aire sous la courbe de charge répartie entre ces deux points.

Relation Entre la Charge Répartie et la Force de Cisaillement

Comprendre la relation entre la charge répartie et la force de cisaillement dans l’analyse structurelle est crucial pour analyser les poutres soumises à divers ...

relation-between-the-distributed-load-and-shear

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Relation Between the Distributed Load and Shear

shear change under distributed loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment.

To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

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relation between the distributed load and shear

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

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key terms and concepts

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | JoVE K12

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Shear Change Under Distributed Loading

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