2-تقرير عن center of pressure

Forces on Submerged Surfaces in Static Fluids
We have seen the following features of statics fluids

• Hydrostatic vertical pressure distribution
  
• Pressures at any equal depths in a continuous fluid are equal
  
• Pressure at a point acts equally in all directions (Pascal's law).
  
• Forces from a fluid on a boundary acts at right angles to that boundary.
  

Objectives:
We will use these to analyse and obtain expressions for the forces on submerged surfaces. In doing this it should also be clear the difference between:

• Pressure which is a scalar quantity whose value is equal in all directions and,
  
• Force, which is a vector quantity having both magnitude and direction.
  

1. Fluid pressure on a surface


Pressure is defined as force per unit area. If a pressure p acts on a small area then the force exerted on that area will be

Since the fluid is at rest the force will act at right-angles to the surface.
General submerged plane
Consider the plane surface shown in the figure below. The total area is made up of many elemental areas. The force on each elemental area is always normal to the surface but, in general, each force is of different magnitude as the pressure usually varies.

We can find the total or resultantforce, R, on the plane by summing up all of the forces on the small elements i.e.

This resultant force will act through the centre of pressure, hence we can say
If the surface is a plane the force can be represented by one single resultant force,
acting at right-angles to the plane through the centre of pressure.
Horizontal submerged plane
For a horizontal plane submerged in a liquid (or a plane experiencing uniform pressure over its surface), the pressure, p, will be equal at all points of the surface. Thus the resultant force will be given by

Curved submerged surface
If the surface is curved, each elemental force will be a different magnitude and in different direction but still normal to the surface of that element. The resultant force can be found by resolving all forces into orthogonal co-ordinate directions to obtain its magnitude and direction. This will always be less than the sum of the individual forces,.
2. Resultant Force and Centre of Pressure on a submerged plane surface in a liquid.



This plane surface is totally submerged in a liquid of density and inclined at an angle of to the horizontal. Taking pressure as zero at the surface and measuring down from the surface, the pressure on an element , submerged a distance z, is given by

and therefore the force on the element is

The resultant force can be found by summing all of these forces i.e.

(assuming and g as constant).
The term is known as the 1^st Moment of Area of the plane PQ about the free surface. It is equal to i.e.

where A is the area of the plane and is the depth (distance from the free surface) to the centroid, G. This can also be written in terms of distance from point O ( as )

Thus:
The resultant force on a plane

This resultant force acts at right angles to the plane through the centre of pressure, C, at a depth D. The moment of R about any point will be equal to the sum of the moments of the forces on all the elements of the plane about the same point. We use this to find the position of the centre of pressure.
It is convenient to take moments about the point where a projection of the plane passes through the surface, point O in the figure.

We can calculate the force on each elemental area:

And the moment of this force is:

are the same for each element, so the total moment is

We know the resultant force from above , which acts through the centre of pressure at C, so

Equating gives,

Thus the position of the centre of pressure along the plane measure from the point O is:

It look a rather difficult formula to calculate - particularly the summation term. Fortunately this term is known as the 2^nd Moment of Area , , of the plane about the axis through O and it can be easily calculated for many common shapes. So, we know:

And as we have also seen that 1^st Moment of area about a line through O,
Thus the position of the centre of pressure along the plane measure from the point O is:

and depth to the centre of pressure is

How do you calculate the 2^nd moment of area?


To calculate the 2^nd moment of area of a plane about an axis through O, we use the parallel axis theorem together with values of the 2^nd moment of area about an axis though the centroid of the shape obtained from tables of geometric properties.
The parallel axis theorem can be written

where is the 2^nd moment of area about an axis though the centroid G of the plane.
Using this we get the following expressions for the position of the centre of pressure

(In the examination the parallel axis theorem and the will be given)
The second moment of area of some common shapes.


The table blow given some examples of the 2^nd moment of area about a line through the centroid of some common shapes.

Shape	Area A	2nd moment of area, , about an axis through the centroid
Rectangle
Triangle
Circle
Semicircle