Hydrostatics: Forces on Surfaces — Worked Examples

A vertical rectangular gate is $2.00\ \text{m}$ wide and $3.00\ \text{m}$ high. Its top edge is $1.00\ \text{m}$ below the water surface. Determine the hydrostatic resultant and the vertical depth of the center of pressure.

A horizontal rectangular hatch measures $1.20\ \text{m}$ by $0.800\ \text{m}$ and lies at a uniform depth of $4.50\ \text{m}$ below an open water surface. Determine the resultant force and its point of application.

A $2.00\ \text{m}$ wide by $3.00\ \text{m}$ long rectangular gate is inclined $60.0^\circ$ above horizontal. The top edge is $1.00\ \text{m}$ vertically below the water surface. Determine the resultant force and center-of-pressure depth.

A circular gate has diameter $1.50\ \text{m}$ and is inclined $60.0^\circ$ above horizontal. Its highest point is $2.00\ \text{m}$ vertically below the water surface. Determine the hydrostatic resultant and vertical center-of-pressure depth.

A vertical triangular plate has a horizontal base $2.00\ \text{m}$ wide at the free surface and an apex $3.00\ \text{m}$ below the surface. Determine the hydrostatic resultant and center-of-pressure depth.

A vertical trapezoidal gate extends from the free surface to a depth of $4.00\ \text{m}$. Its width increases linearly from $1.00\ \text{m}$ at the top to $3.00\ \text{m}$ at the bottom. Determine the hydrostatic resultant and center-of-pressure depth.

A vertical rectangular plate is $1.50\ \text{m}$ wide and $2.00\ \text{m}$ high. Its top edge is $1.00\ \text{m}$ below an open water surface. Use a pressure-diagram decomposition to determine the resultant and its depth.

A vertical gate is $2.00\ \text{m}$ wide and $2.00\ \text{m}$ high, with its top hinge at the water surface. A horizontal cable attached at the bottom holds it closed. Neglect gate weight. Determine cable tension.

A quarter-cylindrical gate of radius $2.00\ \text{m}$ and width $1.00\ \text{m}$ forms the curved boundary of a quarter-circle water region. The circle center lies at the intersection of a vertical wall and the free surface, so the gate extends from the free surface to a depth of $2.00\ \text{m}$. Water is on the concave side of the gate. Determine the horizontal, vertical, and resultant hydrostatic forces on the gate.

At a particular depth in a thin cylindrical tank, the internal liquid pressure exceeds the external pressure by $250\ \text{kPa}$. The tank inside diameter is $1.20\ \text{m}$ and wall thickness is $8.00\ \text{mm}$. Using the thin-wall membrane model, determine hoop tension per unit axial length and local hoop stress.

A simplified gravity structure has weight $1200\ \text{kN}$ acting $2.00\ \text{m}$ upstream of the downstream toe. The horizontal hydrostatic resultant is $300\ \text{kN}$ acting $1.50\ \text{m}$ above the base. Uplift is represented by a $200\ \text{kN}$ upward resultant acting $2.50\ \text{m}$ upstream of the toe. If the base-friction coefficient used in the simplified sliding model is $\mu=0.65$, calculate the overturning safety ratio and the friction-only sliding safety ratio. Do not judge acceptability without a specified design criterion.