In this lesson, we will derive equations for

determining the location of the center of pressure on a plane surface. The tank on the right contains a liquid which

is exposed to the ambient pressure. We focus on a section of wall of arbitrary

shape and want to determine the location where the resultant force, FR, acts. This location is called the center of pressure. Knowledge of the location of the center of

pressure is important in situations where a gate may swing open or a retaining wall

may tip over. The origin of the coordinate system is shown

on the left diagram. The x-coordinate comes out of the screen and

is oriented along the free surface. The y-coordinate is oriented along the section

of wall we wish to examine. The coordinates of the center of pressure

are (xR, yR). The centroid of the surface is labeled as

C and its coordinates are (xC, yC) The location of some arbitrary area dA on

the surface is located at coordinate (x, y). We first will determine the y-coordinate of

the center of pressure, yR. The net hydrostatic pressure force at some

small area dA is gamma times y sin(theta) times dA. The moment this force produces about the x-axis

is the moment arm y times dFnet. Integrating over the entire surface gives

us the total moment produced by the hydrostatic pressure force. yR is the moment arm needed for the resultant

force, FR, to produce the same moment as this integral. In other words, the resultant force FR and

center of pressure location yR produce an equivalent moment on the wall as the original

pressure field. The resultant force FR is equal to gamma times

sin(theta) yC times A. We plug in the expression for FR and dFnet

into the equation, and pull gamma and sin(theta) out of the integral because they are constant. We are left with the integral of y-squared

dA. This quantity is the second moment of area

about the x-axis, and we will denote this as Ixx,O. So the right side of the equation becomes

gamma sin(theta) times Ixx,O. It can be difficult to calculate the second

moment of area about an arbitrary x-axis, but for common shapes it is easy to calculate

the second moment of area about a parallel axis that passes through the centroid, which

we will call Ixx,C. The parallel axis theorem relates Ixx,O and

Ixx,C. Plugging in the expression for Ixx,O, we find

that yR is equal to yC plus Ixx,C divided by yC times A. The second term, Ixx,C divided by yC times

A, is the distance between the centroid and center of pressure along the y-axis. We can follow a similar procedure to find

the x-coordinate of the center of pressure, xR. The equation for the net force exerted on

area dA is the same, gamma times y sin(theta) dA. But now we want to know the moment produced

by dFnet about the y-axis. The moment arm for dFnet is x, and the moment

produced by dFnet is x times dFnet. We can find the total moment about the y-axis

by integrating x dFnet over the entire wall. xR is the moment arm needed for the resultant

force, FR, to produce the same moment about the y-axis as the entire pressure field. Plug in the expression for FR and dFnet, and

pull gamma and sin(theta) out of the integral on the right because they are constant. The integral of x y dA is the product moment

of area with respect to the x and y axes, and is denoted by Ixy,O. So the right side of the equation becomes

gamma sin(theta) times Ixy,O. It can be difficult to calculate the product

moment of area about an arbitrary set of x-y axes. However, for common shapes it is easy to calculate

this quantity about a parallel set of orthogonal axes that pass through the centroid. We will call this quantity Ixy,C. The parallel axis theorem relates Ixy,O and

Ixy,C. Plugging in the expression for Ixy,O, we find

that xR is equal to xC plus Ixy,C divided by yC A. The second term, Ixy,C divided by yC A, is

the distance between the centroid and center of pressure along the x-axis. For many problems in fluid mechanics, it is

not necessary to calculate xR. Although not derived in this video, for closed

tanks that are pressurized to a gage pressure P0, the equations for xR and yR need to be

modified slightly. xR becomes xC plus Ixy,C divided by the quantity

yC plus P0 divided by gamma sin(theta), times A. yR becomes yC plus Ixx,C divided by the quantity

yC plus P0 divided by gamma sin(theta), times A. We now will list the equations for the area,

Ixx,C, Ixy,C, and the location of the centroid for common shapes. For rectangular walls, the area is the height

times the base. Ixx,C is 1/12 times the height to the third

power times the base. Ixy,C is 0 since the shape is symmetric about

the y-axis. The centroid is located at the midpoint between

the left and right sides, and the midpoint between the top and bottom sides. For triangular walls, the area is the 1/2

times the height times the base. Ixx,C is the height to the third power times

the base, divided by 36. Ixy,C is one divided by 72, times the base,

times the height squared, times the quantity base minus two times d, where the distance

from the vertex to the left side of the triangle, as measured along the x-axis. If the triangle is symmetric about the y-axis,

Ixy,C becomes 0. The distance from the base to the centroid

is one-third times the height. The distance from the left side of the triangle

to the centroid is one-third base plus the distance d. For circular walls, the area is pi times the

radius squared. Ixx,C is 1/4 pi times the radius to the fourth

power. Ixy,C is 0 since the shape is symmetric about

the y-axis. The centroid is located at the center of the

disk. For semi-circular walls, the area is one-half

pi times the radius squared. Ixx,C is 0.1098 times the radius to the fourth

power. Ixy,C is 0 since the shape is symmetric about

the y-axis. The centroid is located at a distance four-thirds

times the radius divided by pi away from the base of the semi-circle. For quarter circle walls, the area is one-quarter

pi times the radius squared. Ixx,C is 0.05488 times the radius to the fourth

power. Ixy,C is -0.01647 times the radius to the

fourth power. The centroid is located at a distance four-thirds

times the radius divided by pi away from the two flat ends of the quarter-circle.