Name: hydrostatic force on dam walls with integration
Uploaded: 2026-01-12T12:24:31.000Z
Duration: 1 min 30 s
Channel: JoVE

La pression hydrostatique provient du poids du fluide au repos et augmente proportionnellement à la profondeur.Pour une surface horizontale immergée à une profondeur fixe, la pression est constante et égale au produit de la densité du fluide, de l’accélération due à la gravité et de la profondeur.Mais pour une surface verticale comme la face d’un barrage, chaque point se trouve à une profondeur différente, ce qui fait augmenter la pression avec la profondeur et crée une distribution des forces non uniforme.Pour calculer la force hydrostatique totale à partir de cette pression variable, la section submergée est divisée en fines bandes horizontales. Chaque bande se trouve à une profondeur spécifique et a une surface égale à sa largeur multipliée par une petite épaisseur verticale.La pression sur chaque bande est égale à la densité de poids du fluide multipliée par sa profondeur sous la surface, et la force sur cette bande est égale à la pression multipliée par la surface.La somme de ces forces sur toutes les bandes approxime la force totale à l’aide d’une somme de Riemann. Lorsque l’épaisseur de la bande tend vers zéro, la somme devient une intégrale définie.Cette intégrale exprime la force hydrostatique totale en pression multipliée par la fonction de largeur, intégrée sur la profondeur submergée.Cette méthode est essentielle en génie civil pour concevoir des structures de soutènement efficaces.

applications de l’intégration au calcul de la force hydrostatique

Applications of Integration to Find Hydrostatic Pressure

La force hydrostatique est la force totale exerc&#xE9;e par un fluide au repos sur une surface. Pour une surface horizontale immerg&#xE9;e &#xE0; une profondeur fixe, la pression est constante et se calcule comme le produit de la masse volumique du fluide, de l&#x2019;acc&#xE9;l&#xE9;ration de la pesanteur et de la profondeur. Dans le cas d&#x2019;un mur de barrage vertical immerg&#xE9; dans l&#x2019;eau, cette force n&#x2019;est pas uniform&#xE9;ment r&#xE9;partie en raison de l&#x2019;augmentation de la pression avec la profondeur. Cette variation r&#xE9;sulte du poids cumul&#xE9; de l&#x2019;eau au-dessus de chaque point. L&#x2019;int&#xE9;gration est utilis&#xE9;e pour prendre en compte la variation continue de la pression le long de la surface immerg&#xE9;e et calculer avec pr&#xE9;cision la force hydrostatique totale.La pression dans un fluide au repos augmente lin&#xE9;airement avec la profondeur. &#xC0; une profondeur h, la pression P est donn&#xE9;e par :\begin{equation*}P=\rho gh\end{equation*}&#xA0;o&#xF9; &#x3C1; est la masse volumique du fluide et g est l&#x2019;acc&#xE9;l&#xE9;ration de la pesanteur.Pour calculer la force totale, la surface immerg&#xE9;e est divis&#xE9;e en bandes horizontales d&#x2019;&#xE9;paisseur infinit&#xE9;simale. Chaque bande situ&#xE9;e &#xE0; la profondeur h poss&#xE8;de une largeur w(h) et une &#xE9;paisseur diff&#xE9;rentielle dh ; son aire est donc w(h)dh. La force &#xE9;l&#xE9;mentaire dF s&#x2019;exer&#xE7;ant sur cette bande est :\begin{equation*}dF=P( \text{area})=(\rho gh)( w(h)dh)\end{equation*}&#xA0;En sommant les forces s&#x2019;exer&#xE7;ant sur toutes ces bandes sur l&#x2019;intervalle de profondeur allant de a &#xE0; b, la force hydrostatique totale F s&#x2019;exprime par l&#x2019;int&#xE9;grale d&#xE9;finie :\begin{equation*}F=\int_a^b\rho gh w(h)dh\end{equation*}Cette formulation prend en compte les variations de largeur du mur du barrage en fonction de la profondeur, permettant ainsi une mod&#xE9;lisation pr&#xE9;cise des surfaces planes et courbes.Cette m&#xE9;thode est essentielle dans des domaines de l&#x2019;ing&#xE9;nierie tels que le g&#xE9;nie civil et le g&#xE9;nie hydraulique. Elle garantit la conception s&#xFB;re et efficace d&#x2019;ouvrages tels que les barrages, les murs de sout&#xE8;nement, les vannes et les r&#xE9;servoirs de stockage, en d&#xE9;terminant l&#x2019;influence de la pression variable du fluide sur l&#x2019;int&#xE9;grit&#xE9; structurelle. La capacit&#xE9; de l&#x2019;int&#xE9;gration &#xE0; rendre compte de la distribution exacte des forces offre aux ing&#xE9;nieurs la pr&#xE9;cision n&#xE9;cessaire &#xE0; la stabilit&#xE9; et &#xE0; la s&#xE9;curit&#xE9; des structures en contact avec un fluide.Le calcul diff&#xE9;rentiel et int&#xE9;gral sous-tend les pratiques modernes de l&#x2019;ing&#xE9;nierie en transformant les lois physiques en expressions int&#xE9;grables. Il permet l&#x2019;analyse et la construction de syst&#xE8;mes complexes, ancr&#xE9;s dans le r&#xE9;el.

Applications de l’intégration au calcul de la force hydrostatique

La force hydrostatique est la force totale exerc&#xE9;e par un fluide au repos sur une surface. Pour une surface horizontale immerg&#xE9;e &#xE0; une ...

applications-of-integration-to-find-hydrostatic-pressure

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Calculus

Calcul

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Application of Techniques of Integration

Application des techniques d'intégration

KEY TERMS AND CONCEPTS

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hydrostatic force on dam walls with integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth.

A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall.

Integration helps measure this changing force. Pressure in a still fluid rises linearly with depth. At depth h, the pressure P depends on the fluid density ρ and gravity g.

To find the total force, the submerged surface is split into very thin horizontal strips. Each strip has width w(h) and small height dh, so its area is w(h)dh. The force on one strip is written as a differential force dF.

The forces on all strips are added with a definite integral from depth a to depth b. This method also works when the wall width changes with depth. It can model both flat and curved surfaces.

These calculations matter in civil and hydraulic engineering. Engineers use them to design dams, retaining walls, gates, and storage tanks safely. Integration gives the precision needed to handle changing fluid pressure and protect structural integrity.

Hydrostatic Force on Dam Walls with Integration

hydrostatic-force-on-dam-walls-with-integration

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applications of integration to find hydrostatic pressure

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

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key terms and concepts

Learn how integration is used to calculate hydrostatic force on dam walls by adding the changing pressure of water across submerged surfaces.

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calculus

Hydrostatic Force on Dam Walls with Integration

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