Hydraulics: Hydrostatic Forces - Engineering Notes PDF

Okay, here is the converted markdown format of the attached document: ## HYDRAULICS ### HYDROSTATIC FORCES ON SURFACES In engineering designs where a liquid is contained by surfaces, such as a dam, the side of a ship, a water tank, or a levee, it is necessary to calculate the forces and their loc...

Okay, here is the converted markdown format of the attached document: ## HYDRAULICS ### HYDROSTATIC FORCES ON SURFACES In engineering designs where a liquid is contained by surfaces, such as a dam, the side of a ship, a water tank, or a levee, it is necessary to calculate the forces and their locations due to the liquid on the various surfaces. The liquid is most often water, but it could also be oil or some other liquid. Consider the general surface shown below. The liquid acts on the plane area shown as a section of the wall; a top view gives additional detail of the geometry. The force on the plane surface is due to the pressure $p = \gamma h$ acting over the area, i.e., $F = \int p \, dA = \gamma \int h \, dA $ $ = \gamma \sin \alpha \int y \, dA = \gamma \bar{y} A \sin \alpha$ where $\bar{y}$ is the distance to the centroid of the plane area; the centroid is identified as the point C. The above equation can also be expressed as $F = \gamma h A$ The image is a diagram that includes: * Free surface: $p=0$ * Inclined plane area * $F$ force vector acting on the inclined plane area * Angle $\alpha$ between the free surface and the inclined plane area. * y is along the inclined plane area * Area dA * $y_p$ * $\gamma h dA$ ### Total Pressure on Plane Surface The total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity. ### Center of Pressure on Plane Surfaces Any plane surface subjected to hydrostatic pressure is acted upon by an infinite number of parallel forces the magnitudes of which vary with the depth, below the free surface, of the various infinitesimal areas on which the respective forces act. These parallel forces may be replaced by a single resultant force $P$ or $F$. The point on the surface at which this resultant force acts is called the center of pressure. If the total hydrostatic pressure on any surface were applied at the center of pressure the same effect would be produced on the surface, consider as a free body, as is produced by the distributed pressure. The position of the horizontal line containing the center of pressure of a plane surface subjected to hydrostatic pressure may be determined by taking moments of all the forces acting on the surface about some horizontal axis in its plane. Let us assume that the force acts at some point called the center of pressure, located by the point ($x_c$, $y_p$). To determine where the force acts, we must recognize that the sum of the moments of all the infinitesimal forces must equal the moment of the resultant force, i.e., $y_p F = \gamma \int_A y h \, dA$ $= \gamma \sin \alpha \int_A y^2 \, dA = \gamma I_x \sin \alpha$ where $I_x$ is the second moment of the area about the x-axis. The parallel-axis transfer theorem states that $I_x = \bar{I} + A \bar{y}^2$ where $\bar{I} $ is the moment of the area about its centroidal axis. So, the expression to solve the location of the center of pressure, $y_p$, is $y_p = \bar{y} + \frac{\bar{I}}{A \bar{y}}$ $e = \frac{\bar{I}}{A \bar{y}}$ where $\bar{y} = \frac{h}{\sin \alpha}$ ### Total Pressure on Curved Surface The image shows a diagram of a curved surface with pressure acting on it. * The curved surface $AB$. * Vertical direction is $y$ * Horizontal direction is $x$ * An infinitesimal area with pressure ($p$) acting on it is shown on curved surface. * $P_x$ is the horizontal force acting on the surface. * $P_y$ is the vertical force acting on the surface. where $F_x$ = total force acting on the vertical projection of the curved surface. $F_y$ = weight of the imaginary or real fluid directly above the curved surface. Note: For cylindrical and spherical surfaces, the total force $F$ always passes through the center of the circle defined by its surface.