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

静水压力由静止流体的重量产生，并随深度成比例增加。对于在固定深度处浸没的水平表面，压强是恒定的，且等于流体密度、重力加速度与深度的乘积。但对于大坝表面等垂直面，每一点所处的深度不同，导致压强随深度增加而增大，从而形成非均匀的力分布。为了计算由这种变化的压力所产生的总静水力，将浸没部分划分为若干细长的水平条带。每一条带位于特定深度，其面积等于宽度乘以一个微小的垂直厚度。每条带上的压强等于流体的重量密度乘以其在表面以下的深度，而该带上的力等于压强乘以面积。对所有条带上的这些力进行求和，可利用黎曼和来近似总力。当条带厚度趋近于零时，该求和变为一个定积分。该积分表达式表示总静水力等于压力乘以宽度函数，并在淹没深度上进行积分。该方法在土木工程中对于设计高效的挡土结构至关重要。

利用积分法计算静水压力产生的合力

Applications of Integration to Find Hydrostatic Pressure

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

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Application of Techniques of Integration

积分技术的应用

KEY TERMS AND CONCEPTS

关键术语与概念

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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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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