Question

The rigid gate, OAB, of Fig. P11.23 is hinged at O and rests against a rigid support at B. What minimum horizontal force, P, is required to hold the gate closed if its width is 3 m? Neglect the weight of the gate and friction in the hinge. The back of the gate is exposed to the atmosphere.

Answer 1

Concepts and reason

Using hydrostatic equation, the hydrostatic forces acting on the gate are determined. Calculate the polar moment of inertia and center of pressure using free body diagram of the gate.

The minimum horizontal force required to hold the gate in closed position is determined by applying equilibrium condition for moments.

Fundamentals

Hydrostatic force:

The liquid pressure which acts perpendicular to the surface forms a linear distributed resultant force is known as hydrostatic force.

The formula to calculate the hydrostatic force is given as follows:

…… (1)

Here, hydrostatic force is , specific weight is , centroid is and area is .

Weight of a body:

It is the force acting in the body due to gravity. It is also defined as the product of mass and acceleration due to gravity.

The weight of the body is expressed as follows:

$W=mxg $

Here, mass of the body is m and acceleration due to gravity is g.

Specific weight:

Specific weight is the ratio of weight to the unit volume. It is denoted by the symbol .

Center of pressure:

Because of internal pressure in the fluid the total resultant force acting at a particular spot is called Center of pressure.

The formula to calculate the center of pressure is as follows:

…… (2)

Here, moment of inertia about gravity is , centroid about horizontal axis is and area is .

Moment of inertia:

The product of area with the square of the distance from the reference axis is known as moment of inertia. It is also known as the second moment of area. It is denoted by and its unit is .

Write the equation of moment of inertia for a rectangle shape.

…… (3)

Here, base of the rectangle is and depth of the rectangle is .

Moment:

It refers to the propensity of the force to cause rotation in a body about any fixed point. The moment’s magnitude can be obtained by multiplying force’s magnitude with the perpendicular distance at which the force acts. The moment is denoted by and its unit is.

Write the formula for moment due to force about any point.

Here, the force is F and the perpendicular distance of force from point is d.

Equilibrium of a rigid body:

An object is said to be in equilibrium when the sum of external forces and couples are zero.

For a rigid body to be in equilibrium in three dimensions, the sum of external forces acting along , and directions have to zero.

For a rigid body to be in equilibrium in three dimensions, the sum of external couples about any point should be zero.

Write the equilibrium condition for moments about any point.

Here, the sum of all moments about the point is .

The free body diagram of the rigid gate is shown as in Figure (1).

Here, hydrostatic forces are and , heights at which the hydrostatic forces and act on the gate are and and horizontal and vertical reactions at point are and .

Calculate the height at which the hydrostatic force acts on the gate.

Calculate the area of the gate .

Here, height of the surface above the point is and height of the gate is .

Substitute for and for .

Calculate the hydrostatic force acting on the gate.

Substitute for , for , and for .

Calculate the height at which the hydrostatic force acts on the gate .

Calculate the area of the gate .

Here, base of the gate is and width of the gate if it hold the gate at closed position is .

Substitute for and for .

Calculate the hydrostatic force acting on the gate.

Substitute for , for and for .

Calculate the moment of inertia of the rigid gate along horizontal direction.

Put for and for .

From Equation (2), calculate the center of pressure.

Substitute for , for , for and for .

For Figure (1), take moment about point .

Substitute for , for , and for .

Ans:

The minimum horizontal force required to hold the gate in closed position is .